UC-NRLF 


I 


I 


CAMBRIDGE   PHYSICAL   SERIES. 


ELECTRICITY 

AND 

MAGNETISM 


HonDon:    C.   J.   CLAY   AND   SONS, 

CAMBRIDGE    UNIVERSITY   PRESS   WAREHOUSE, 

AVE   MARIA  LANE, 

AND 

H.  K.  LEWIS, 
136,  GOWER  STREET,  W.C. 


©laggofo:    50,  WELLINGTON  STREET. 

Eetpjtg:    F.  A.   BROCKHAUS. 

$efo  ?|orfe:   THE  MACMILLAN  COMPANY. 

Bombao  anti  Calrutta:   MACMILLAN  AND  CO.,  LTD. 


[All  Rights  reserved.] 


ELECTEICITY 


AND 


MAGNETISM 

AN    ELEMENTARY    TEXT-BOOK 
THEORETICAL   AND   PRACTICAL 


BY 


R.   T.   GLAZEBROOK,   M.A,   F.R.S. 

v 

DIRECTOR    OP    THK     NATIONAL     PHYSICAL    LABORATORY, 
FELLOW    OF    TRINITY   COLLEGE,    CAMBRIDGE. 


CAMBRIDGE : 

AT    THE    UNIVERSITY    PRESS 
1903 


NERAL 

yK  ( C 


Catnbritige : 

PEINTKl)   BY   J.    AND   C.    F.    CLAY 
AT   THE    UNIVERSITY   PRESS. 


ELECTRICITY. 


G.  E.  1 


PREFACE. 

SOME  words  are  perhaps  necessary  to  explain  the 
publication  of  another  book  dealing  with  Elementary 
Electricity.  A  considerable  portion  of  the  present  work 
has  been  in  type  for  a  long  time  ;  it  was  used  originally 
as  a  part  of  the  practical  work  in  Physics  for  Medical 
Students  at  the  Cavendish  Laboratory  in  connexion  with 
my  lectures,  and  was  expanded  by  Mr  Wilberforce  and 
Mr  Fitzpatrick  in  one  of  their  Laboratory  Note-books  of 
Practical  Physics. 

When  I  ceased  to  deliver  the  first  year  course  I  was 
asked  to  print  my  lectures  for  the  use,  primarily,  of  the 
Students  attending  the  practical  classes  ;  the  lectures  on 
Mechanics,  Heat  and  Light  have  been  in  type  for  some 
years.  Other  claims  on  my  time  have  prevented  the  issue 
of  the  present  volume  until  now,  when  it  appears  in 
response  to  the  promise  made  several  years  ago. 

Meanwhile  the  subject  has  changed ;  but  while  this  is 
the  case  the  elementary  laws  and  measurements  on  which 
the  science  is  based  remain  unaltered,  and  I  trust  the 
book  may  be  found  of  service  to  others  besides  my 
successors  at  the  Cavendish  Laboratory. 

As  in  the  other  books  of  the  Series,  I  have  again  to 
thank  Mr  Fitzpatrick  for  his  very  valuable  assistance. 


VI  PREFACE 

He  has  read  all  the  proofs  and  suggested  numerous 
improvements,  and  has  thus  brought  the  book  up  to  date 
as  representing  a  course  which  many  years'  experience  has 
proved  to  be  a  useful  one  for  elementary  students. 

The  book  is  to  be  used  in  the  same  way  as  its  pre- 
decessors. The  apparatus  for  most  of  the  Experiments  is 
of  a  simple  character  and  can  be  supplied  at  no  great 
expense  in  considerable  quantities. 

Thus  the  Experiments  should  all,  as  far  as  possible, 
be  carried  out  by  the  members  of  the  class,  the  teacher 
should  base  his  reasoning  on  the  results  actually  obtained 
by  his  pupils.  Ten  or  twelve  years  ago  this  method  was 
far  from  common ;  the  importance  to  a  School  of  a 
Physical  Laboratory  is  now  more  generally  recognized ; 
it  is  with  the  hope  that  the  book  may  be  of  value  to 
those  who  are  endeavouring  to  put  the  method  in  practice 
that  it  is  issued  now. 

R.   T.   GLAZEBROOK. 


NATIONAL  PHYSICAL  LABORATORY. 
July  19,  1903. 


CONTENTS. 

CHAP.  PAGE 

I.  ELECTROSTATICS  ;  FUNDAMENTAL  FACTS  ...  3 

II.  ELECTRICITY  AS  A  MEASURABLE  QUANTITY      .        .  19 

III.  MEASUREMENT  OF  ELECTRIC  FORCE  AND  POTENTIAL  44 

IV.  CONDENSERS 58 

V.      ELECTRICAL  MACHINES 70 

VI.  MEASUREMENT  OF  POTENTIAL  AND  ELECTRIC  FORCE  87 

VII.  MAGNETIC  ATTRACTION  AND  REPULSION          .        .  105 
VIII.    LAWS  OF  MAGNETIC  FORCE 122 

IX.      EXPERIMENTS  WITH  MAGNETS 126 

X.      MAGNETIC  CALCULATIONS 135 

XI.      MAGNETIC  MEASUREMENTS 157 

XII.     TERRESTRIAL  MAGNETISM 171 

XIII.  THE  ELECTRIC  CURRENT 182 

XIV.  RELATION    BETWEEN    ELECTROMOTIVE    FORCE    AND 

CURRENT 210 

XV.  MEASUREMENT  OF  CURRENT     .        .  223 


Vlll  CONTENTS 

N 

CHAP.  PAGE 

XVI.     MEASUREMENT  OF  RESISTANCE  AND  ELECTROMOTIVE 

FORCE    .        .        *        .        .        ...        .  240 

XVII.     MEASUREMENT  OP  QUANTITY  OP  ELECTRICITY,  CON- 
DENSERS        .        .        .        ....        .  279 

XVIII.    THERMAL  ACTION  OF  A  CURRENT    ....  286 

XIX.  THE  VOLTAIC  CELL.     (THEORY.)      ....  302 

XX.  ELECTROMAGNETISM 323 

XXI.  MAGNETISATION  OF  IRON  ....                .  341 

XXII.  ELECTROMAGNETIC  INSTRUMENTS      .        .        .        .361 

XXIII.  ELECTROMAGNETIC  INDUCTION 373 

XXIV.  APPLICATIONS  OF  ELECTROMAGNETIC  INDUCTION      .  388 
XXV.     TELEGRAPHY  AND  TELEPHONY         .        .                .  406 

XXVI.  ELECTRIC  WAVES      .        .        .        .        .        .        .411 

XXVII.  TRANSFERENCE  OP  ELECTRICITY  THROUGH  GASES  ; 

CORPUSCLES  AND  ELECTRONS        ....  418 

ANSWERS  TO  EXAMPLES  .        .        .     '    .        .        .  431 

INDEX  435 


CHAPTER   I. 


ELECTROSTATICS;    FUNDAMENTAL   FACTS. 


1.  Electric  Attraction.  The  word  electricity  is 
derived  from  -tjXcKTpov  the  Greek  for  amber  ;  if  we  take  a 
piece  of  sealing-wax  or  a  glass  rod  and  rub  it  with  dry 
flannel  or  silk  it  will  be  found  to  have  acquired  the  power  of 
attracting  light  objects  such  as  bits  of  tissue  paper  or  feathers. 
This  property  of  attraction  was  first  discovered  by  the  Greeks 
in  amber;  hence  when  Dr  Gilbert  about  1600  found  that 
numerous  other  substances  possessed  it  when  rubbed  he  called 
them  all  "Electrics." 

This  attraction  may  be  illustrated  in  various  ways.  Take 
a  dry  glass  rod  and  rub  it  with  a  dry  silk  handkerchief,  then 
hold  it  over  a  number  of  bits 
of  tissue  paper ;  the  tissue 
paper  jumps  up  to  the  rod ; 
probably  some  bits  stick  to 
the  rod  while  others  after 
touching  it  are  repelled  and 
fly  away.  The  same  effect  can 
be  shewn  by  rubbing  a  piece 
of  sealing-wax  or  a  rod  of 
ebonite  with  dry  flannel  or 
catskin. 

We  can  however  study 
the  effects  better  thus.  Sus- 
pend a  light  pith  ball  or  a 
small  bit  of  feather  by  a  thin 
silk  thread  ;  then  rub  a  glass 
rod  with  silk  and  bring  it  Fig.  1. 

1—2 


4 


ELECTRICITY 


[CH.  I 


near  the  ball ;  the  ball  is  attracted  to  the  glass  as  in 
Figure  1 ;  allow  the  two  to  come  into  contact,  the  ball  is 
then  repelled  from  the  glass. 

Again  take  a  second  pith  ball  suspended  in  the  same  way 
and  rub  a  rod  of  ebonite  with  dry  flannel ;  then  bring 
the  rod  near  the  ball,  it  is  first  attracted  and  then  after 
contact  is  repelled  from  the  rod. 

The  glass  and  the  ebonite  have  both  been  electrified  by 
friction,  the  first  ball  has  been  in  contact  with  the  glass,  the 
second  with  the  ebonite ;  the  glass  and  the  ebonite  are  both 
said  to  be  electrified.  Each  has  the  power  of  attracting  to 
itself  a  light  body  such  as  the  pith  "ball  and  then  repelling 
it.  A  body  which  has  been  electrified  is  said  to  be  charged 
with  electricity  or  to  have  an  electric  charge. 

2.  Two  kinds  of  electrification.  Many  other 
substances  can  be  electrified  by  friction,  such  as  fur,  wool, 
ivory,  sulphur  and  india-rubber,  but  the  states  of  electrification 
produced  in  these  various  bodies  are  not  the  same;  we  can 
shew  in  fact  that  there  are  two  kinds  of  electrification. 

EXPERIMENT  1.  To  shew  tJiat  there  are  two  kinds  of 
electrification  produced  by  friction. 

A  wire  stirrup  is  suspended  by  means  of  a  fine  silk  fibre 
as  shewn  in  Figure  2.  The  stirrup  is  intended  to  support 


Fig.  2. 


1-3]  ELECTROSTATICS;    FUNDAMENTAL   FACTS  5 

a  rod  of  glass  or  of  ebonite  after  it  has  been  electrified.  Rub 
one  end  of  an  ebonite  rod  with  a  piece  of  dry  flannel  and 
support  it  in  the  stirrup  so  that  it  hangs  horizontally.  Elec- 
trify in  the  same  manner  one  end  of  a  second  ebonite  rod 
and  bring  it  near  the  electrified  end  of  the  suspended  rod  ; 
the  latter  is  repelled.  Now  electrify  a  glass  rod  by  rubbing 
it  with  silk  and  bring  it  near  the  electrified  end  of  the 
suspended  ebonite  rod ;  the  ebonite  rod  is  attracted.  Thus 
while  two  ebonite  rods  when  electrified  repel  each  other,  an 
ebonite  rod  and  a  glass  one  attract ;  the  effects  on  the 
suspended  ebonite  rod  of  the  glass  and  ebonite  are  opposite ; 
we  may  express  this  by  saying  that  the  two  states  of  electri- 
fication are  of  opposite  sign  ;  it  is  usual  to  call  the  electricity 
produced  on  glass  by  friction  with  silk  positive,  that  produced 
on  ebonite  or  sealing-wax  when  rubbed  with  wool  or  with 
catskin  is  called  negative;  these  signs  are  matters  of  conven- 
tion, the  electricity  on  the  glass  might  have  been  called 
negative,  that  on  the  ebonite  positive,  had  this  convention 
been  agreed  to  instead  of  that  which  has  actually  been 
adopted. 

v  Thus  we  have  seen  from  the  above  experiment  that  there 
are  two  opposite  states  of  electrification,  and  further  that  two 
bodies  similarly  electrified  repel  each  other,  while  two  bodies 
oppositely  electrified  attract  each  other. 

We  could  vary  the  above  experiment  by  electrifying  a 
piece  of  glass  by  silk  and  supporting  it  in  place  of  the 
ebonite ;  we  should  then  find  that  it  was  attracted  by  elec- 
trified ebonite,  repelled  by  electrified  glass. 

Moreover  if  the  silk  and  the  catskin  be  very  dry  we  can 
shew  in  the  same  way  that  they  also  are  electrified  by  friction, 
for  on  holding  the  catskin  near  to  the  suspended  glass  it  is 
attracted  while  the  silk  which  was  used  to  rub  the  glass  is 
repelled.  Thus  the  catskin  is  positively  electrified  by  rubbing 
ebonite,  and  the  silk  negatively  by  rubbing  glass. 

3.  Conductors  and  non-conductors.  If  an  elec- 
trified piece  of  ebonite  be  laid  on  another  piece  of  ebonite 
or  on  a  sheet  of  dry  glass  or  a  piece  of  dry  silk,  it  will  retain 
its  electrification  for  some  time  ;  if  however  it  be  rubbed 
gently  with  a  damp  cotton  cloth  or  with  the  hand  or  touched 


6  ELECTRICITY  [CH.  I 

all  over  with  a  piece  of  tin-foil  or  other  metal,  it  at  once 
loses  its  electrification.  The  ebonite,  the  glass,  and  the  silk 
are  said  to  be  non-conductors  of  electricity  ;  the  damp  cloth, 
the  hand,  and  the  metal  are  called  conductors  of  electricity. 
A  conductor  allows  the  free  passage  of  electricity  over  its 
surface ;  if  we  hold  a  piece  of  brass  tube  in  the  hand 
and  rub  it  with  catskin  or  flannel,  no  sign  of  electrification 
is  produced  ;  if  however  we  fix  the  brass  tube  on  to  an 
ebonite  rod  and  rub  it  with  dry  catskin  taking  care  to  avoid 
touching  the  brass  with  anything  but  the  fur,  we  find  that 
the  brass  is  negatively  electrified  ;  if  brought  near  an  ebonite 
rod  suspended  as  in  Experiment  1  it  will  repel  it.  The  brass 
tube,  the  human  body,  arid  the  earth  are  all  conductors; 
some  electricity  is  produced  by  the  friction  in  the  first  case, 
but  it  spreads  itself  over  the  brass  and  passes  through  the 
body  of  the  experimenter  to  the  earth  ;  in  the  second  case 
also  electricity  is  produced  in  the  brass  but  is  prevented 
from  passing  to  the  earth  by  the  ebonite  handle.  Thus  the 
tube  remains  electrified. 

There  is  this  difference  however  between  it  and  the  ebonite ; 
that  part  only  of  the  ebonite  which  has  been  rubbed  is  elec- 
trified, while  the  whole  of  the  brass  tube  shews  signs  of 
electrification.  Electrification  produced  by  friction  on  a  non- 
conductor remains  where  it  was  produced,  while  on  a  conductor 
it  spreads  itself  over  the  whole  surface. 

This  can  be  shewn  in  the  following  manner : — 

On  dusting  a  body  with  a  mixture  of  powdered  red  lead 
and  yellow  sulphur  which  has  been  well  shaken  together,  both 
powders  become  electrified  by  the  friction,  the  lead  being 
positive  and  the  sulphur  negative  ;  if  the  body  be  unelectrified 
both  powders  can  be  easily  removed  by  blowing  on  the  surface ; 
if  the  body  be  positively  electrified  the  sulphur  is  held  by  it 
and  the  lead  is  removed  by  blowing,  the  surface  becomes  yellow 
when  blown  upon.  If  on  the  other  hand  the  body  be 
negatively  electrified,  the  red  lead  is  retained  and  the  sulphur 
blown  off,  the  surface  becomes  red ;  we  have  thus  an  easy 
test  of  the  nature  of  the  electrification  of  a  body. 

Now  support  a  rod  of  ebonite  and  a  brass  tube,  each  in 
some  non-conducting  support.  Rub  one  end  of  the  ebonite 


3-5]        ELECTROSTATICS;   FUNDAMENTAL  FACTS  7 

with  catskin  and  dust  it  with  the  red  lead-sulphur  mixture ; 
the  end  which  was  rubbed  becomes  red,  neither  powder  adheres 
to  the  other  end,  the  ebonite  is  only  electrified  where  it  was 
rubbed.  Repeat  the  experiment  with  the  brass  tube,  it  be- 
comes red  all  over,  the  electrification  has  distributed  itself  all 
over  the  brass. =H__— 

DEFINITION.      A^  subsj^nce    which   is    a   non-conductor   o/*T~N 
electricity  is  said  to  be  an^  Ingula^or  or  jto  insulate,  r  and^  a  | 
Jwdy  supported  by  an  insulator  is  said  to  be  insidafed- , 

4.  Properties  of  Insulators.     Bodies  can  be  divided 
roughly  into   two    classes,   conductors,   which   allow  the  free 
transference    of    electricity    from    point    to    point,    and    non- 
conductors   or  insulators    in   which    that    transference    takes 
place  very  slowly  indeed  if  at  all.      A  perfect  insulator  would 
be  a  body  which  entirely  stopped  the  passage  of  electricity ; 
substances  ordinarily  classed  as  insulators,  glass,  shellac,  silk, 
etc.,  are  not  perfect,  but  while  in  concluctors  the  transference 
is   so    rapid    as    to    be    practically    instantaneous,   in   a  good 
insulator  it  is  very  slow.     Indeed  glass  when  thoroughly  dry, 
fused  quartz,  paraffin  wax  and  various  other  substances  are 
practically  perfect  insulators.     Water  as  we  know  it  is  a  good 
conductor,  though    it   appears   probable   that   perfectly  pure 
water  would  be  an  insulator.     Air  and  the  other  permanent 
gases   when   dry  are  insulators,    cotton    thread    especially   if 
slightly  damp  is  a  conductor,  silk  is  an  insulator. 

5.  Electrification  by  induction.    Bring  an  electrified 
ebonite  rod  near  to  a  conductor  supported  011  an  insulating 
stand,  and  dust  the  conductor  with  the  red  lead  and  sulphur 
powder  as  described  in  Section  3  ;  on  blowing  on  the  conductor 
the  end  near  the  ebonite  will  become  yellow,  the  other  end  red; 
the  nearer  end  has  become  positively  electrified,  the  further 
end  negatively.      On  removing  the  ebonite  the-  electrification 
disappears.     The  conductor   is  said    to   be  electrified   by  in- 
duction.     If  a  positive   body  had   been   used  instead  of  the 
ebonite  to  produce  the  induction,  the  nearer  end  would  have 
been  negative,  the  further  end  positive. 


8  ELECTRICITY  [CH.  I 

In  the  above  experiment  the  state  of  induction  lasts  only 
so  long  as  the  charged  body  is  near  the  conductor.  We  can 
however  produce  by  induction  a  permanent  state  of  electri- 
fication thus. 

EXPERIMENT  2.     To  electrify  a^conductor  by  induction. 

Bring  an  electrified  ebonite  rod  near  to  an  insulated  con- 
ductor, and  while  the  ebonite  is  in  position  touch  the  conductor 
for  a  moment  with  the  finger.  Then  remove  the  ebonite ;  on 
dusting  the  insulated  conductor  with  the  lead-sulphur  powder 
and  blowing  on  it,  it  will  become  yellow  shewing  that  it  is 
positively  electrified  ;  if  a  positively  charged  body  had  been 
used  instead  of  the  ebonite  to  produce  the  induction,  the 
insulated  conductor  would  at  the  end  of  the  experiment 
remain  negatively  electrified,  it  would  thus  appear  red.  More- 
over in  both  these  cases  there  is  attraction  between  the  charged 
rod  and  the  insulated  conductor. 

To  shew  this,  charge  one  end  of  the  ebonite  rod  and 
suspend  it  in  the  stirrup  as  in  Experiment  1.  Then  bring  up 
near  to  it  the  insulated  conductor  on  its  stand,  the  charged 
end  of  the  rod  is  attracted.  If  the  conductor  is  not  insulated 
the  attraction  is  more  marked ;  this  can  be  shewn  by  touching 
the  insulated  conductor  with  the  hand  when  near  the  rod, 
the  rod  is  drawn  still  closer  to  the  conductor. 

We  have  already  seen  that  on  the  insulated  conductor 
there  is  positive  electrification  near  the  ebonite,  and  negative 
at  a  distance  from  it ;  this  enables  us  to  explain  the  attrac- 
tion, for  we  may  imagine  the  conductor  divided  into  two 
parts,  one  of  these  is  positively  electrified,  the  other  negatively. 
There  is  attraction  between  the  ebonite  and  the  positive  part, 
repulsion  between  the  ebonite  and  the  negative  part.  But  the 
distance  between  the  first  two  being  less  than  that  between  the 
second  two,  the  attraction  is  greater  than  the  repulsion  and 
hence  on  the  whole  the  ebonite  is  attracted  to  the  conductor. 

6.     Explanation    of   electrical    attraction.      We 

may  use  these   results  to  explain  the  phenomena  of   attrac- 
tion and  repulsion  described  in  the  first  section,  thus  : 

The  bits  of  paper,  the  feather,  and  the  pith  balls  are  all 
of  them  conductors.  When  the  charged  ebonite  rod  is  brought 


5-7] 


ELECTROSTATICS  ;    FUNDAMENTAL   FACTS 


9 


near  the  pith  ball  electric  induction  takes  place,  the  ball 
becoming  positively  electrified  on  its  side  nearest  the  ebonite 
and  negatively  electrified  on  the  opposite  side ;  the  ball  is  thus  ' 
attracted  by  the  ebonite ;  on  contact  the  positive  electrification 
of  the  ball  is  discharged  by  combination  with  some  of  the 
negative  electrification  of  the  ebonite,  the  induced  negative 
charge  remains  on  the  ball ;  hence  the  ball  and  the  ebonite 
being  similarly  charged  repulsion  takes  place  and  the  ball  flies 
away  from  the  ebonite. 

7.   Electroscopes.  An  instrument  for  detecting  whetherj) 
a  body  is  electrified  or  not  is  called  an  electroscope. 

In  an  electroscope  as  a  rule  the  attraction  or  repulsion 
between  two  electrified  bodies  is  made  use  of  to  determine  the 
presence  of  electrification..  Thus  the  pith  ball  suspended  by 
a  silk  fibre  might  be  used  as  an  electroscope ;  a  more  sensitive 
electroscope  however  can  be  constructed  as  is  shewn  in  Figure  3. 


Fig.  s. 


Fig.  4. 


Two  pith  balls  are  connected  by  a  linen  or  cotton  thread 
and  suspended  from  a  glass  support.  Cotton  or  linen  is  used 
because  it  is  a  conductor  so  that  any  charge  communicated  to 


10  ELECTRICITY  [CH.  I 

either  ball  is  shared  by  the  other.  If  the  two  balls  become 
electrified  they  repel  each  other  and  stand  apart.  A  number 
of  observations  might  be  made  with  such  an  electroscope. 
The  gold-leaf  electroscope  however,  Figure  4,  which  acts  on 
the  same  principle,  is  much  more  delicate,  the  balls  are 
replaced  by  two  strips  of  gold  or  aluminium  leaf.  These 
hang  from  a  metal  rod  inside  a  glass  vessel.  The  rod  is 
inserted  into  a  glass  tube  which  is  well  coated  with  shellac 
varnish  and  passes  through  a  cork  or  stopper  in  the  neck  of 
the  vessel.  The  shellac  improves  the  insulating  properties 
of  the  glass  greatly1.  The  metal  rod  terminates  in  a  metal 
plate  or  knob  which  is  thus  insulated  but  is  in  connexion 
with  the  gold  leaves.  For  accurate  work  the  surface  of 
the  glass  vessel  is  partly  covered  with  strips  of  tin-foil, 
spaces  being  left  between  the  strips  through  which  the  gold 
leaves  can  be  seen,  or  in  some  cases  a  cylinder  or  cage  of 
wire  gauze  is  placed  inside  the  vessel.  The  object  of  these 
precautions  is  to  prevent  electrification  on  the  surface  of  the 
glass  or  on  external  objects  from  influencing  the  gold  leaves 
directly. 

If  the  plate  or  knob  becomes  electrified  by  any  means  the 
gold  leaves  will  also  become  charged  with  like  electricity  and 
will  repel  each  other ;  the  distance  to  which  they  stand  apart 
will  clearly  depend  in  some  way  on  their  state  of  electrification 
and  may  be  utilized  to  determine  whether  the  original  source 
of  the  electrification  is  strong  or  weak. 

We  can  perform  a  number  of  experiments  with  the  gold- 
leaf  electroscope  to  illustrate  the  matters  already  referred  to 
as  well  as  some  of  the  fundamental  laws  of  the  subject. 

IXPERIMENT  3.  To  determine  whether  a  body  is  electrified 
and  to  shew  that  electrification  is  produced  by  friction. 

Iring  a  body  such  as  a  rod  of  ebonite  or  sealing-wax  near 
the  knob  of  an  electroscope  and  observe  what  happens ;  if  the 
gold  leaves  do  not  diverge  the  body  is  not  electrified,  if  they 
do,  the  body  is  charged  ;  in  this  case  discharge  the  body  either 

1  For  delicate  work  fused  quartz  is  now  sometimes  used  as  the 
insulator  instead  of  the  glass  rod. 


7]  ELECTROSTATICS  ;    FUNDAMENTAL   FACTS  11 

by  drawing  it  across  your  hand  or  by  passing  it  through  a  gas 
flame.  Then  rub  the  rod  with  a  piece  of  dry  flannel  or  fur. 
On  again  bringing  it  near  the  electroscope  the  leaves  diverge, 
the  body  is  electrified ;  if  the  fur  is  dry  and  is  fastened  to 
the  end  of  an  insulating  rod  and  not  held  in  the  hand,  it  may 
be  possible  to  shew  that  it  is  also  electrified  by  the  friction, 
by  removing  the  ebonite  and  bringing  the  fur  near,  when  the 
leaves  again  diverge. 

This  last  experiment  does  not  always  succeed  because  the  damp  hand 
is  a  conductor  and  much  of  the  electrification  of  the  fur  has  escaped  in 
the  handling. 

EXPERIMENT  4.      To  shew  that  there  are  two  kinds  of  electri- 
fication. 

This  has  already  been  shewn  in  Experiment  2. 

Rub  an  ebonite  rod  with  flannel  and  bring  it  near  an 
electroscope,  the  leaves  diverge ;  remove  the  ebonite  rod  and 
bring  near  a  glass  rod  which  has  been  rubbed  with  dry  silk,  the 
leaves  again  diverge.  Repeat  the  experiment  with  the  ebonite 
rod,  but  while  it  is  near  the  electroscope  bring  up  the  glass  rod, 
note  what  happens  ;  as  the  glass  rod  is  brought  near,  the  leaves 
begin  to  fall  together  again  ;  the  electrification  of  the  glass 
rod  reduces  the  divergence  due  to  that  of  the  ebonite  rod,  the 
two  states  of  electrification  ,  are  opposite ;  if  the  glass  be 
brought  sufficiently  near,  the  leaves  may  collapse  entirely  and 
then  begin  to  diverge  again ;  if  this  is  the  case  on  removing 
the  ebonite  rod  they  will  diverge  still  more. 

The  electrification  of  the  glass  rod  is  said  to  be  vitreous  or 
positive,  that  on  the  ebonite  resinous  or  negative. 

EXPERIMENT  5.  To  charge  an  electroscope  (a)  by  conduction, 
(b)  by  induction. 

(a)  Electrify  the  ebonite  rod  by  friction  with  a  piece  of 
dry  flannel,  it  will  be  negatively  electrified  ;  allow  it  to  touch 
the  knob  of  the  electroscope,  the  leaves  diverge  and  remain 
divergent  when  the  ebonite  is  removed,  though  less  widely 
than  when  the  rod  was  in  contact.  The  rod  has  given  up  part 
of  its  negative  charge  to  the  electroscope  which  has  thus  been 
negatively  electrified  by  conduction.  Discharge  the  electro- 
scope by  touching  the  knob  for  a  moment  with  the  finger ; 


12  ELECTRICITY  [CH.  I 

on  again  bringing  up  the  ebonite  the  leaves  again  diverge, 
shewing  that  only  a  part  of  the  charge  was  in  the  first  instance 
communicated  to  the  electroscope. 

(b)  Electrify  the  ebonite  again  by  friction  and  bring  it 
near  the  knob  of  the  electroscope,  the  leaves  diverge.  When 
the  ebonite  is  in  position  touch  the  knob  for  a  moment  with 
your  finger,  the  leaves  collapse  ;  remove  your  finger  and  then 
remove  the  ebonite  rod;  the  leaves  again  diverge;  the  electro- 
scope is  charged  but  in  this  case  the  ebonite  has  not  given  up 
any  of  its  electrification,  the  electroscope  has  been  charged  by 
induction.  Moreover  the  charge  in  this  case  is  positive.  For 
we  have  seen  that  a  negatively  charged  body  induces  positive 
electrification  in  the  near  parts  of  any  neighbouring  con- 
ductor and  repels  negative  electrification  to  a  distance.  Tims 
in  the  first  part  of  the  experiment  the  knob  is  positively 
electrified  by  induction,  the  leaves  negatively.  When  the 
electroscope  is  touched  it  becomes  electrically  part  of  the  earth. 
The  knob  is  still  positively  charged  but  the  negative  electri- 
fication passes  to  the  earth.  On  removing  the  hand  and  then 
the  ebonite  this  positive  electrification  is  free  to  spread  itself 
over  the  gold  leaves,  which  thus  diverge. 

We  may  shew  experimentally  that  the  final  electrification 
is  negative  in  (a),  positive  in  (6),  thus  :  electrify  a  glass  rod  by 
friction  with  silk,  and  bring  it  near  the  electroscope.  In  case 

(a)  it  will  be  noted  that  the  leaves  collapse  as  the  glass  rod  is 
brought  near,  they  are"  therefore  negatively  electrified  ;  in  case 

(b)  they  diverge  still  further,  they  are  positively  electrified. 

EXPERIMENT  6.  To  illustrate  the  difference  between  con- 
ductors and  non-conductors  and  to  charge  a  conductor  by 
friction. 


Charge  the  electroscope  and  then  touch  the  knob  \vith 
various  substances  held  in  the  hand,  taking  care  that  none  of 
them  are  previously  electrified,  and  note  the  effects.  With 
some  substances  the  electroscope  is  practically  unaffected,  these 
are  insulators  ;  with  others  the  leaves  immediately  collapse, 
these  are  good  conductors  ;  with  others  again  there  is  a  slow 
fall  of  the  leaves,  these  last  substances  are  bad  conductors. 

Again  take  a  brass  tube  on  an  ebonite  handle  and  holding 


7]  ELECTROSTATICS;  FUNDAMENTAL  FACTS  13 

the  brass  in  the  hand  rub  it  with  flannel  taking  care  not  to 
rub  the  ebonite.  On  bringing  the  brass  up  to  an  uncharged 
electroscope  no  effect  is  produced,  the  brass  tube  is  not 
electrified.  Now  hold  the  brass  by  the  ebonite  handle  and 
again  rub  it  with  the  flannel,  on  bringing  it  up  to  the  electro- 
scope the  leaves  diverge,  the  brass  has  been  electrified  by 
friction,  and  the  electrification  has  been  prevented  from 
escaping  by  the  ebonite  handle. 


EXPERIMENT  7.  To  test  wliether  a  charge  is  positive  or 
negative. 

Charge  the  electroscope  with  positive  electricity  by  in- 
duction as  in  Experiment  5.  Electrify  a  glass  rod  by  friction 
with  silk  and  bring  it  near  the  knob  of  the  electroscope. 
Observe  that  the  leaves  diverge  further,  the  positive  electrifi- 
cation of  the  glass  repels  more  positive  electricity  into  the 
leaves. 

Electrify  an  ebonite  rod  by  friction  with  flannel  and  bring 

it  near  the  knob.     At  first  the  divergence  of  the  leaves  is 

decreased,  and  as  the  rod  is  brought  nearer,  if  it  be  strongly 

-I  electrified,  they  collapse  entirely,  on  bringing  the  rod  nearer 

still  they  diverge  again,  but  this  time  with  negative  electricity. 

It  is  therefore  necessary  to'  observe  the  first  indication  of 
the  electroscope. 

/  Thus,  starting  with  an  electroscope  positively  charged,  if 
the  approach  of  a  body  causes  still  further  divergence  the 
body  is  positively  charged,  if  it  causes  the  leaves  to  collapse 
the  body  is  negatively  charged. 

EXPERIMENT  8.  To  shew  that  both  kinds  of  electricity  are 
produced  simultaneously  (a)  by  friction  or  (b)  by  induction. 

We  have  already  proved  this  by  means  of  experiments 
with  the  red  lead-sulphur  powder;  in  the  following  experiments 
the  gold-leaf  electroscope  is  used. 

(a)  Tie  a  piece  of  dry  flannel  or  of  catskin  on  to  an 
ebonite  rod,  and  having  charged  an  electroscope  positively 
rub  the  flannel  on  a  second  ebonite  rod.  On  bringing  this 
ebonite  near  to  the  electroscope  the  leaves  collapse,  it  is 
negatively  charged  ;  on  removing  the  ebonite  and  bringing  the 


14  ELECTRICITY  [CH.  I 

flannel  near  they  open  more  widely  than  before,  the  flannel  is 
positive. 

The  flannel  is  tied  to  the  ebonite  so  that  it  need  not  be 
handled,  the  ebonite  is  a  much  better  insulator  than  the 
flannel. 

(6)  Obtain  two  equal  metal  balls  A  and  B  mounted 
on  insulating  stands  and  place  them  eight  or  ten  centimetres 
apart.  Connect  them  by  a  piece  of  wire  held  in  an  insulating 
handle  so  that  the  connexion  between  the  two  can  be  broken 
without  touching  either  with  the  hand.  Bring  a  body  charged 
positively  up  near  the  ball  A.  Then  remove  the  connecting  wire 
arid  examine  the  electrifications  of  A  and  B  by  bringing  each  in 
its  turn  up  to  a  charged  electroscope.  In  doing  this  care 
must  be  taken  to  handle  only  the  insulating  stands  of  the 
balls.  It  will  be  found  that  A  is  charged  negatively,  B 
positively.  If  the  inducing  body  had  been  negative  then  A 
would  be  positive,  B  negative.  The  two  opposite  electrifi- 
cations are  produced  by  induction. 

A  block  of  paraffin  wax  makes  a  very  good  insulating 
stand  for  this  and  similar  experiments. 

EXPERIMENT  9.      To  prove  there  is  no  electrification  inside 
/'  .  a  hollow  closed  conductor  provided  there  is  no  insulated  charged 
body  within  it. 

For  the  hollow  closed  conductor  in  this  experiment,  we 
use  a  tall  metal  can  15  to  20  cm.  in  height,  which  should  be 
well  insulated  by  being  placed  on  a  block  of  clean  paraffin. 
The  can  is  not  completely  closed  but  with  the  accuracy  to 
which  we  can  work  the  error  due  to  this  is  negligible.  A 
small  brass  ball  about  a  centimetre  in  diameter,  supported 
either  by  an  ebonite  handle  or  by  a  silk  thread,  is  also 
wanted. 

Place  the  conductor  on  an  insulating  support  and  charge 
the  ball  either  by  induction  or  by  the  use  of  some  electrical 
machine  (see  Section  49).  Verify  that  the  ball  is  electrified 
by  bringing  it  near  to  the  electroscope.  Place  the  ball  inside 
the*  conductor  and  let  it  touch  the  side,  then  lift  it  out  and 
aga;n  bring  it  near  the  electroscope,  the  leaves  remain 
undisturbed,  the  ball  has  lost  its  charge  by  contact  with 


7-8]        ELECTROSTATICS;  FUNDAMENTAL  FACTS  15 

the  interior  of  the  conductor.  Repeat  this  several  times, 
charging  the  ball  and  then  allowing  it  to  touch  the  interior 
of  the  conductor ;  on  lifting  it  out  the  ball  is  in  all  cases 
completely  discharged  ;  its  electrification  has  passed  to  the 
exterior  of  the  conductor.  Verify  that  the  exterior  of  the 
conductor  is  electrified  either  by  touching  it  on  the  exterior 
with  the  ball  and  then  bringing  the  ball  near  to  the  electro- 
scope when  the  leaves  diverge,  or  by  bringing  the  conductor 
near  the  electroscope— if  this  last  method  be  adopted,  the 
conducting  part  of  the  vessel  must  not  be  handled. 

Thus  whenever  an  electrified  conductor'  is  placed    inside  ' 
a  hollow  conductor  and  allowed  to  touch  it,  the  electrification 
passes  at  once  to  the  exterior  of  the  hollow  conductor,  even 
though  that  be  already  charged.     There  is  no  electrification 
inside. 

We  have  supposed  throughout  this  experiment  that  there 
is  no  second  electrified  body  within  the  hollow  conductor  but 
insulated  from  it.  If  we  had  a  second  charged  ball  and 
held  it  inside  the  conductor  without  allowing  the  two  to  touch, 
we  should  find  on  repeating  the  experiment  that  the  first  ball 
was  not  completely  discharged  by  contact  with  the  conductor. 

EXPERIMENT  10.  To  prove  that  there  is  no  electrical  force 
inside  a  hollow  closed  conductor  provided  there  is  no  charged 
insulated  body  within. 

Make  a  cage  of  fine  wire  gauze  or  thin  sheet  metal  to 
surround  the  gold-leaf  electroscope  entirely,  if  sheet  metal 
is  used  pierce  in  it  two  or  three  holes  through  which  the 
gold  leaves  can  be  observed.  Charge  the  cage  by  the  aid 
of  an  electrical  machine  or  in  any  other  way,  the  gold  leaves 
remain  undisturbed.  There  is  no  force  inside  ;  if,  however, 
a  charged  body  be  introduced  into  the  cage  the  leaves  diverge 
and  indicate  that  there  is  electric  force.  The  metal  cage  entirely 
screens  the  electroscope  from  all  external  electrical  force.  There 
is  no  force  inside  due  to  external  electrification. 

We  see  hence  one  reason  for  surrounding  the  gold  leaves  with  gauze 
or  \vith  strips  of  tin-  foil  as  described  in  Section  7. 

8.  The  Proof  Plane.  A  proof  plane,  Fig.  5,  is  a  useful 
piece  of  apparatus  by  which  the  state  of  electrification  of  the 


16 


ELECTRICITY 


[CH.  I 


surface  of  a  body  can  be  examined.     It  consists  of  a  disc  of 

metal  curved  so  as  to  fit  the  surface  of 

the  body  approximately  and  supported 

by  an  insulating  handle.     When    the 

proof  plane  is  placed  against  the  charged 

body  it  becomes  practically  part  of  the 

surface  of  that  body.     The  electricity 

from  the  surface  under  the  proof  plane 

passes  to  the  outer  surface  of  the  proof 

plane  itself.     On   removing  the  proof 

plane  this  electricity  is  removed  with 

it,  and  thus  the  state  of  electrification 

of  any  portion  of  the  surface  can  be 

examined.  Fig.  5. 

9.     No  electrification  within  a  closed  conductor. 

The  results  of  the  two  last  Experiments  are  of  great 
theoretical  importance,  and  various  other  experiments  have 
been  devised  to  illustrate  them. 

Thus  in  Fig.  6  A  represents  a  sphere  suspended  by  an  in- 
sulating thread.  B  and  C  are  two  hollow  hemispheres  which  fit 
it  closely  and  have  insulating  handles  attached.  Electrify  A, 


8-9] 


ELECTROSTATICS;   FUNDAMENTAL  FACTS 


17 


B  and  C  being  unelectritied.  Then  tit  the  two  hemispheres 
together  over  A,  remove  them  and  bring  them  to  the  electro- 
scope, they  will  be  found  to  be  electrified.  On  testing  A  it 
will  be  seen  that  it  has  completely  lost  its  electrification. 

Faraday's  butterfly  net  experiment  is  illustrated  in  Figure  7. 


Fig.  7. 

A  muslin  net,  a  butterfly  net  attached  to  a  vertical  wire, 
is  held  in  an  insulating  stand.  A  silk  thread  is  attached  to 
the  vertex  of  the  net,  and  by  pulling  this  either  surface  of  the 
muslin  can  be  brought  to  the  outside. 

Charge  the  net  with  electricity  and  examine  its  surfaces 
by  aid  of  a  small  proof  plane,  touching  the  surfaces  in  turn 
and  bringing  the  plane  after  each  contact  up  to  the  electro- 
scope. It  will  be  found  that  the  outer  surface  is  charged, 
the  inner  uncharged.  Now  invert  the  net  by  pulling  the 
loose  end  of  the  thread  so  that  the  inner  surface  becomes  the 
outer.  The  electricity  is  on  the  new  outer  surface ;  the  surface 
which  was  originally  uncharged  has  passed  to  the  outside 
and  has  become  charged,  and  vice  versd. 

In  another  experiment  Faraday  constructed  a  large  metal 
box.  He  placed  inside  this  box  the  most  delicate  electroscopes 
he  possessed  and  went  inside  it  himself.  The  outside  of  the 

G.  E.  2 


18  ELECTRICITY  [CH.  I 

box  was  then  electrified  as  strongly  as  possible,  but  he  was  not 
able  to  discover  by  the  most  delicate  means  in  his  power  any 
trace  of  electrification  inside. 

We  have  already  seen  that  two  bodies  similarly  electrified 
repel  each  other,  we  have  not  discussed  what  the  law  of  force 
between  them  is.  Now  this  law  can  be  deduced  mathe- 
matically from  the  experimental  fact  that  there  is  no  force 
inside  a  hollow  closed  conductor,  and  hence  we  see  the  theo- 
retical importance  of  Faraday's  experiments.  See  Section  26. 


CHAPTER  II. 

ELECTRICITY  AS  A   MEASURABLE  QUANTITY. 

1O.  Quantity  of  Electricity.  Hitherto  we  have 
spoken  in  general  terms  about  electrifying  a  body  or  charging 
it  with  electricity,  and  we  have  seen  that  both  kinds  of 
electrification,  positive  and  negative,  are  produced  simul- 
taneously. The  experiments  moreover  have  been  qualitative, 
we  have  not  attempted  any  quantitative  measurements.  We 
are  now  about  to  describe  experiments  which  shew  us  that 
we  can  look  upon  an  electrical  charge  as  a  quantity  which 
like  other  quantities  can  be  measured  in  proper  units,  and 
that  we  may  speak  of  charging  a  body  with  a  definite  quantity 
of  electricity  just  as  we  speak  of  pouring  a  definite  quantity 
of  water  into  a  pail.  While  furthermore,  not  only  are  the 
opposite  electricities  always  produced  simultaneously,  but  they 
are  produced  in  equal  quantities. 

EXPERIMENT  11.  To  justify  the  use  of  the  term  Quantity 
of  Electricity.  Faraday's  ice-pail  experiments. 

Place  a  tall  narrow  metal  vessel — the  can  used  in  Ex- 
periment 9  will  do  well — on  an  insulating  stand  and  connect  it 
as  in  Figure  8  to  the  electroscope.  A  second  smaller  vessel, 
in  Figure  9,  of  similar  shape  but  with  an  insulating  handle 
and  a  number  of  metal  balls  fastened  to  silk  threads  or 
insulating  handles  will  be  wanted. 

Charge  one  of  the  metal  balls  and  introduce  it  into  the 
metal  conductor,  taking  care  that  the  two  do  not  touch.  The 

2—2 


20 


ELECTRICITY 


[CH.  II 


electroscope  leaves  diverge,  and  it  may  be  shewn  by  a  test 
that  their  electrification  is  of  the  same  sign  as  that  of  the 
charged  ball.  Move  the  ball  about  inside  the  conductor, 
then  provided  it  is  kept  some  way  from  the  mouth  it  will 
be  found  that  the  divergence  of  the  leaves  remains  the  same 
however  the  position  of  the  ball  is  altered. 


Fig.  8. 


Fig.  9. 


Introduce  now  the  smaller  vessel  as  well  as  the  ball  inside 
the  conductor,  taking  care  to  remove  previously  any  charge  it 
may  possess.  The  divergence  of  the  leaves  remains  the  same. 
Place  the  ball  inside  the  smaller  vessel  and  let  the  two  touch. 
Still  the  leaves  remain  divergent  exactly  as  before.  Remove 
the  ball,  it  will  be  found  to  be  unelectrified  and  the  leaves 
diverge  as  before.  Introduce  some  uncharged  balls  carried  by 
insulating  threads  within  the  insulated  conductor,  the  leaves 
remain  apart ;  allow  all  the  conductors  to  touch  the  interior 
of  the  vessel,  this  does  not  affect  the  leaves ;  then  remove 
them  all,  they  will  be  found  to  be  discharged  and  the  leaves 
remain  as  before. 


10]  ELECTRICITY    AS    A    MEASURABLE   QUANTITY  21 

Now  all  these  various  results  are  consistent  with  the 
statement  that  originally  we  put  a  definite  quantity  of 
electricity  inside  the  tall  vessel,  and  that  the  effect  on  the 
electroscope  depends  solely  upon  that  quantity.  By  our 
various  operations  we  have  altered  the  distribution  of  that 
quantity,  nothing  that  we  have  done  has  changed  its  amount. 
By  the  last  operation  we  transferred  the  electricity,  still 
unchanged  in  quantity,  from  the  inside  to  the  outside  of 
the  vessel. 

By  electrifying  a  conductor  we  have  produced  some  change 
in  it  which  remains  unaltered  in  amount  as  measured  by  the 
effect  on  the  gold  leaves  unless  it  is  allowed  to  come  into 
contact  with  another  conductor.  So  long  as  the  first  conductor 
is  insulated  the  charged  state  remains  constant  in  quantity. 

The  experiments  just  described  and  others  which  follow  were  first 
performed  by  Faraday  using  ice-pails  for  the  tall  closed  conductors,  hence 
the  name  now  given  to  them. 


) 


EXPERIMENT  12.  To  shew  that  the  induced  charge  is  equal 
and  opposite  to  the  inducing  charge  when  the  conductor  in  which 
induction  is  produced  completely  surrounds  the  charged  body. 

Connect  up  the  larger  "ice-pail"  again  to  the  electroscope. 
Charge  one  of  the  balls  with  positive  electricity  suppose,  and 
introduce  it  into  the  "  ice-pail."  The  leaves  of  the  electroscope 
diverge  with  positive  electricity.  Note  their  divergence,  then 
touch  the  ice-pail  with  the  finger,  the  leaves  collapse,  the 
external  electrification  has  escaped  to  the  earth.  Remove 
the  finger,  and  afterwards  remove  the  ball.  The  leaves 
again  diverge  and  to  the  same  extent  as  previously,  but 
their  charge  is  negative. 

Thus  by  introducing  the  positively  electrified  ball  negative 
electrification  is  induced  on  the  inside  of  the  ice-pail  and  an 
equal  positive  charge  is  driven  to  the  outside  and  to  the 
leaves.  When  the  pail  is  touched,  this  positive  charge  passes 
to  the  earth  and  the  leaves  collapse,  but  on  removing  the 
ball  the  induced  negative  charge  distributes  itself  over  the 
exterior  surface. 

Moreover  this  negative  electricity  is  equal  in  amount  to 
the  original  positive  charge. 


22  ELECTRICITY  [CH.  II 

For  introduce  again  the  charged  ball,  the  leaves  collapse ; 
then  allow  the  ball  to  touch  the  inside  and  remove  it.  The 
ball  and  the  "ice-pail"  are  both  discharged,  hence  the  negative 
charge  on  the  "ice-pail"  was  equal  and  opposite  to  the  positive 
charge  on  the  ball. 

EXPERIMENT  13.  To  determine  if  t/ie  charges  on  two  bodies 
are  equal. 

Place  one  ball  within  the  ice-pail  and  note  the  divergence 
of  the  leaves,  remove  it  and  replace  it  by  the  second,  if  the 
leaves  again  diverge  to  an  equal  extent,  the  two  bodies  have 
equal  charges. 

Some  difficulty  is  introduced  into  all  these  experiments  by  leakage,  it 
is  necessary  for  success  that  the  insulation  should  be  good  and  that  the 
operations  should  be  rapidly  performed. 

With  a  view  to  rendering  the  gold-leaf  electroscope  more  useful  for 
quantitative  experiments,  a  scale  is  sometimes  fixed  behind  the  leaves. 
The  amount  of  divergence  can  then  be  noted  on  the  scale  and  it  can  be 
seen  more  readily  whether  the  divergence  in  two  different  experiments  is 
tbe  same  or  not. 

We  may  use  the  ice-pail  arrangements  for  some  other 
experiments. 

EXPERIMENT  14.  To  prove  that  equal  and  opposite  quan- 
tities of  electricity  are  produced  by  friction. 

This  is  practically  a  repetition  of  Experiment  4. 

Fasten  a  piece  of  fur  on  to  an  ebonite  handle  and  rub 
another  ebonite  rod  with  the  fur,  taking  care  that  all  the 
apparatus  is  originally  free  from  charge. 

>  On  introducing  either  the  fur  or  the  ebonite  into  the 
"ice  pail"  the  leaves  diverge.  Introduce  the  two  together. 
1,"^  divergence  is  observed,  the  charges  on  the  fur  and  the 
ebonite  are  equal  and  opposite. 

Or  again,  to  prove  that  the  quantities  of  positive  and  negative 
electricity  produced  by  induction  are  equal  and  opposite,  we  may 
repeat  Experiment  12. 

That  experiment  has  shewn  that  the  two  balls  are  oppositely  charged. 
Place  the  two  simultaneously  within  the  "  ice-pail "  no  divergence  will  be 
observed,  hence  the  quantity  within  the  ice-pail  is  zero,  the  two  charges 
are  equal  in  amount. 


10-12]       ELECTRICITY    AS   A    MEASURABLE   QUANTITY       23 

11.  Theories    of  Electrical   Action.     It   appears 
from  these  and  similar  experiments  that  it  is  quite   reason- 
able to  speak  of  a  Quantity  of  Electricity  and  to  suppose 
that  there  are  two  opposite  kinds  of  electricity  which  we  call 
positive  and  negative.      Now  we  have  seen  already  that  two 
similarly  electrified  bodies,  or  as  we  may  now  say  two  bodies 
charged  with  quantities  of  similar  electricity,  repel  each  other  ; 
we  can  observe,  and  in  some  cases  measure,  the  force  of  repul- 
sion between  the  bodies.     But  it  is  quite  consistent  with,  and 
will  enable  us  to  coordinate,  the  results  of  observations   to 
suppose  that  the  force  acts  between   the  like  quantities  of 
electricity.     The  force  observed  between  the  charged  bodies 
may  really  be  the  resultant  of   the  forces  between  the  like 
electrifications  on  those    bodies ;    the    electrical   charges   are 
attached  to  the  bodies,  and  the  action  between  the  charges 
may  shew  itself  in  a  motion  of  the  bodies.     We  will  suppose  • 
therefore  that  there  is  a  force  of  repulsion  between  two  like 
charges  and  one  of  attraction  between  two  unlike  charges. 

Let  us  now  further  suppose  that  an  uncharged  body 
contains  equal  quantities  of  positive  and  negative  electricity ; 
these  in  the  undisturbed  state  exactly  neutralize  each  other, 
and  the  body  produces  no  electrical  effects ;  if  we  can  in  any 
way  remove  some  of  the  negative  electricity  the  body  remains 
positively  charged  and  vice  versd. 

12.  Explanation  of  Electrostatic  Actions.     We 

do  not  know  what  electricity  is,  how  it  is  connected  with 
matter,  or  what  constitutes  a  state  of  electrification.  We 
can  however  from  the  above  hypotheses  give  a  consistent 
explanation  of  many  electrical  phenomena. 

For  example,  bring  the  body  near  a  positively  charged 
body,  the  positive  electricity  in  the  first  body  is  repelled 
and  the  negative  attracted  by  this  charge,  thus  the  body 
becomes  electrified  by  induction  ;  if  we  can  divide  it  into 
two  parts  as  in  the  case  of  the  two  balls  in  Experiment  12 
we  obtain  two  bodies,  one  charged  positively,  the  other 
negatively. 

Or  again,  when  electrification  is  produced  by  the  friction, 
say,  of  silk  on  glass,  we  must  suppose  that  the  two  electricities 
are  separated  by  the  friction,  and  that  the  glass  retains  more 


24  ELECTRICITY  [CH.  II 

than  its  normal  share  of  the  positive,  the  silk  more  than  its 
normal  share  of  the  negative. 

13.     Electrical   Distribution.     Surface   Density. 

If  we  are  to  look  upon  the  electrification  of  a  body  as  the 
distribution  of  electricity  over  the  surface  of  the  body,  we 
may  ask  the  question,  According  to  what  law  does  the 
electricity  distribute  itself  1  Will  there  be  the  same  quantity 
on  each  unit  of  area  of  the  surface  or  will  this  quantity  vary 
from  point  to  point?  The  mathematical  theory  of  electricity 
gives  us  an  answer  to  this.  It  can  be  shewn  that  we  may 
treat  the  positive  electricity  as  though  it  were  a  fluid  free  to 
move  over  the  surface  of  the  body;  we  must  suppose  the 
particles  of  this  fluid  to  repel  each  other  according  to  a 
certain  law  depending  on  their  distance  apart,  and  calculate 
from  this  what  the  distribution  will  be.  The  negative  elec- 
tricity must  then  be  treated  as  a  similar  fluid  of  mutually 
repulsive  particles,  and  we  must  further  assume  that  there  is 
an  attractive  force  between  the  particles  of  the  two  kinds  of 
fluid. 

The  two-fluid  theory  of  electricity  is  based  on  these 
assumptions,  and  from  them,  the  law  of  force  being  known,  the 
distribution  of  electricity  in  a  number  of  actual  cases  has  been 
calculated. 

But  without  going  into  any  such  elaborate  calculations  as 
would  be  involved  in  the  above,  the  fact  that  in  general  the 
distribution  of  electricity  on  a  charged  conductor  is  not 
uniform  but  depends  on  the  shape  of  a  conductor  and  its 
position  relative  to  other  conductors  can  be  shewn  by  ex- 
periment. 

DEFINITION.  The  quantity  of  electricity  on  each  unit  of( 
area — one  square  centimetre — of  a  charged  conductor  is  known 
\as  the  Surface  Density  of  the  distribution.  -.._.  ' 

It  is  here  implicitly  assumed  that  the  distribution  is  uniform  over 
each  square  centimetre.  In  the  case  of  a  non-uniform  distribution  we 
may  define  the  surface  density  at  any  point  to  be  the  ratio  of  the  charge 
on  any  small  area  containing  the  point  to  the  area  when  it  is  taken  to  be 
so  small  that  the  distribution  over  it  is  uniform.  Cf.  the  difference 
between  uniform  and  variable  velocity,  Dynamics,  Section  22. 


12-13]      ELECTRICITY    AS   A    MEASURABLE   QUANTITY        25 

Now  we  have  seen  that  a  proof  plane  when  applied  to 
a  conductor  and  removed  takes  with  it  the  charge  which 
occupied  the  portion  of  the  conductor  covered  by  the  proof 
plane.  If  then  we  have  a  proof  plane  one  square  centimetre 
in  area  and  apply  it  at  different  points  of  the  surface  the 
charge  removed  will  in  each  case  measure  the  surface  density 
about  that  point.  Now  the  equality  of  these  charges  can  be 
determined  by  the  "  ice-pail "  experiment,  and  though  it  is  not 
possible  numerically  to  compare  the  charges  by  comparing  the 
divergence  of  the  leaves  in  the  different  experiments,  yet  we 
may  say  with  certainty  that  a  large  divergence  implies  a  large 
charge  and  vice  versd. 

If  for  the  electroscope  some  form  of  electrometer,  Sections 
60,  62,  be  substituted  the  charges  can  actually  be  measured, 
and  hence  the  densities  at  different  points  can  be  compared. 

In  this  manner  it  has  been  shewn  for  example  that  on  a 
sphere  the  density  is  uniform,  on  an  elongated  conductor  it  is 
greatest  near  the  ends  ;  when  a  conductor  is  brought  near  a 
positively  charged  body  it  is  found  to  become  negative  near 
the  body,  positive  far  away.  A  line  along  which  there  is  no 
electrification  may  be  drawn  round  the  body.  Again,  it  may 
be  shewn  that  the  density  is  always  great  near  sharp  points 
or  corners.  These  should  in  most  cases  therefore  be  avoided 
in  electrical  apparatus.  Figure  10  shews  in  a  graphical 


Fig.  10. 

manner  the  distribution  of  the  electricity  on  two  conductors ; 
the  distance  between  the  thick  line,  representing  the  body,  and 
the  dotted  line  which  surrounds  it  is  supposed  to  represent 
the  density  ;  when  the  dotted  line  lies  inside  the  body  the 
charge  is  negative,  when  it  is  outside  the  charge  is  positive. 


26  ELECTRICITY  [CH.  II 

The  theory  we  have  just  been  describing  is  often  spoken  of  as  the 
two-fluid  theory  of  electricity ;  if  we  adopt  it  provisionally  we  must 
guard  ourselves  against  looking  upon  electricity  as  a  fluid.  All  that  the 
theory  states  is  that  an  electric  charge  distributes  itself  over  the  surface 
which  separates  a  conductor  from  the  insulating  material  about  it, 
according  to  the  same  laws  as  a  fluid  consisting  of  mutually  repulsive 
particles  would  do  were  it  free  to  move  over  that  surface.  The  question 
whether  the  electric  charge  resides  on  the  surface  of  the  conductor  or  on 
that  of  the  insulating  medium  by  which  it  is  surrounded  is  one  of  great 
importance,  to  which  we  shall  return  later.  See  Section  46. 

14.  Electrical  Pressure  or  Potential.  Consider 
two  insulated  conductors,  A  and  £,  two  metal  balls  suppose, 
supported  by  silk  threads.  Let  A  be  charged  and  B  un- 
charged. Allow  them  to  touch  and  then  separate  them. 
We  can  easily  shew  by  experiment  that  electricity  has  passed 
from  A  to  B.  Let  us  ask  ourselves  the  question  why  has  it 
so  passed,  and  what  limits  or  regulates  the  flow  ^ 

Or,  again,  take  two  charged  bodies  and  compare  the  charges 
on  the  two  by  placing  each  in  turn  in  the  "ice-pail,"  and 
observing  the  effect.  Allow  them  to  touch  and  separate  them. 
On  again  comparing  their  charges  it  will  probably  be  found 
that  the  charge  on  each  has  altered  though  the  total  charge 
remains  the  same.  Electricity  has  again  passed  from  the  one 
ball  to  the  other  and  we  may  ask  the  same  question. 

The  flow  depends  to  some  extent  on  the  charge  on  each 
ball  but  not  on  it  alone,  for  we  can  easily  arrange  the 
experiment  so  that  the  ball  with  the  larger  charge  gains 
electricity  and  that  with  the  smaller  charge  loses  it. 

We  can  obtain  an  answer  to  our  question  most  readily  by 
considering  cognate  problems  in  other  sciences. 

Thus  consider  two  reservoirs  containing  water,  A  and  £, 
Figure  1 1 ,  connected  by  a  tube  having  a  stopcock,  and  suppose 
the  level  of  the  water  in  A  is  higher  than  that  in  B.  On 
opening  the  stopcock  the  water  runs  from  A  to  B.  The 
lower  reservoir  B  may  contain  the*  most  water,  the  flow  is  not 
regulated  by  this  but  by  the  difference  in  level  of  the  water 
surfaces  in  the  two,  or,  and  this  comes  to  the  same  thing,  by 
the  difference  of  pressure  between  the  two  ends  of  the  pipe. 
If  we  suppose,  in  order  to  get  rid  of  the  effect  of  the  weight  of 
the  water  in  the  pipe,  that  the  pipe  is  horizontal  the  water 


13-14]      ELECTRICITY   AS   A   MEASURABLE    QUANTITY        27 

will  flow  through  the  pipe  from  the  end  at  which  the  pressure 
is  greatest  to  that  at  which  it  is  least. 


: 

'.- 
,'. 

^ 

, 
< 

- 

', 

^7777^T77T7\ 

-  

A 

B 

/.•.  ••/////.    '/////.'////////.     ////V 

'ffi^^T^^&T/TTTTTTTTT?/ 

Fig.  11. 

Or,  again,  take  two  gas-bags  filled  with  compressed  air ; 
connect  them  by  an  india-rubber  tube  closed  with  a  stopcock. 
If  the  pressure  of  the  air  is  the  same  in  the  two,  on  opening 
the  stopcock  there  is  no  flow.  If  the  pressure  be  different 
the  flow  is  from  the  bag  at  high  pressure  to  that  at  low 
pressure. 

The  Science  of  Heat  affords  another  illustration1.  When 
two  hot  bodies  are  put  into  thermal  communication  heat  flows 
from  the  body  at  a  high  temperature  to  that  at  a  low  tem- 
perature. 

Moreover  in  these  two  cases  the  flow  of  fluid  continues 
until  the  pressures  in  the  two  reservoirs  are  equalized,  while 
the  flow  of  heat  continues  until  the  temperatures  of  the  two 
bodies  become  the  same. 

Temperature  is  defined  to  be  the  condition  of  a  body  on 
which  its  power  of  communicating  heat  to  or  receiving  heat 
from  other  bodies  depends. 

Corresponding  to  Pressure  in  Hydrostatics  and  to  Tem- 
perature in  the  Science  of  Heat  we  have  in  dealing  with 
Electricity  to  consider  the  Electrical  Pressure  or  Elec- 
trical Potential,  as  it  is  called. 

1  Glazebrook,  Heat,  §  11. 


28 


ELECTRICITY 


[CH.  II 


Electricity  passes  from  a  body  A  to  a  body  B  because  the 
electrical  potential  of  A  is  higher  than  that  of  B  and  the  flow 
goes  on  until  the  potentials  are  equalized  :  the  potential  of  A 
falls,  that  of  B  rises,  and  the  flow  depends  on  the  difference  of 
potential. 

15.     Explanation  of  Electrical  Potentij 


The  Electrical  Potential  of  a  body  measures  the  con-\* 
dition  of  the  body  on  which  its  power  of  communicating\ 
Electricity  to,  or  receiving  Electricity  from,  other  bodies  L 


when  two  bodies  A  and  B  are  put 
connexion  the  direction  of  the  electric  force  is  such  that 
electricity  passes  from  A  to  B,  then  A  is  said  to  be  at  a  higher 
potential  than  B.  Two  bodies  A  and  B  have  the  same 
potential  if  when  they  are  put  into  electrical  communication 
there  is  no  transference  of  electricity  between  them. 

These  statements  should  be  compared  with  the  corre- 
sponding Definition  of  Temperature1. 

It  must  also  be  noted  that  the  above  statement  does  not 
give  us  a  means  of  measuring  Electrical  Potential,  it  gives  a 
name  to  a  quality  of  an  electrified  body  on  which  certain 
important  properties  depend.  One  important  consequence  of 
the  statement  is  the  following^ 

PROPOSITION   1.     All  points  on  a  conductor  in  a  state  of 


electrical  equilibrium  are  at  the  same  potential. 

3r  consider  two  such  points  A  and 
be  any  difference  of  potential  be- 
tween" them  electricity  will  pass 
until  this  difference  is  neutralized, 
the  condition  that  the  electricity 
should  be  in  equilibrium  on  the 
conductor  is  that  the  potential 
should  be  the  same  at  all  points. 

In  the  case  of  the  Earth,  which 
is  a  large  conductor,  differences  of 
potential  do  exist  between  different  points,  and  in  consequence 

1  Heat,  Section  11. 


14-17]      ELECTRICITY   AS   A    MEASURABLE   QUANTITY        29 

currents,  earth  currents,  are  continually  passing,  but  these  are 
small,  and  we  may  treat  the  potential  of  the  Earth  for  most 
purposes  in  the  neighbourhood  of  any  point  of  observation  as 
being  uniform. 

16.  Zero  of  Potential.     The  Earth  is  so  large  that 
any  charge  we  can  give  it  is  insufficient  to  change  its  potential 
appreciably,  just  as  the  ocean  is  so  big  that  the  rain  it  receives 
cannot  appreciably  affect  its  level.     Just  then  as  we  take  as 
our  zero  level  from  which  to  measure  heights  the  mean  level  of 
the  sea  and  treat  it  as  fixed,  so  we  take  as  our  zero,  from  which 
to  measure  potential  or  electrical  level,  the  potential  of  the 
Earth. 

Thus  the  potential  of  the  Earth  is  the  zero  of  electrical 
potential. 

If  positive  electricity  flows  from  a  body  to  the  Earth  that 
body  has  a  positive  potential ;  if  positive  electricity  flows 
from  the  Earth  to  the  body  the  body  has  a  negative 
potential. 

17.  Analogy  between  Pressure,  Temperature, 
and    Potential.     We   have    seen    that   there  is  a  certain 
analogy  between  potential  and  pressure  or  temperature ;  there 
is  however  one  important  point  of  difference  to  be  noted. 

VThe  pressure  in  a  reservoir  of  gas  does  not  depend  in  any 
way  on  the  pressure  in  neighbouring  reservoirs  ;  if  we  have  a 
numHeFof  independent  vessels  filled  with  gas  we  may  alter 
the  pressure  in  any  one  without  affecting  that  of  the  rest. 

Again,  in  the  case  of  heat,  if  we  could  prevent  radiation, 
the  temperature  of  a  number  of  isolated  bodies  would  not  be 
altered  by  bringing  a  hot  body  near ;  in  reality  in  consequence 
of  radiation  heat  is  transferred  in  such  a  case  and  the  tem- 
jratures  of  all  the  bodies  are  gradually  changed. 

In  the  case  j)f  electricity  however  the  potential  of  any 
conductor  depends  011  that  of  all  the  other  conductors  in  its 
neighbourhood ;  if  we  bring  a  charged  body  near  a  number  of 
other  charged  bodies  the  potential  of  each  body  in  the  system 
is  very  rapidly,  practically  immediately,  altered,  the  change 
bakes  place  not  by  a  slow  gradual  process  like  the  absorption 
radiation  but  at  once. 


30  ELECTRICITY"-  [CH.  II 

The  facts  of  electrical  induction  shew  that  this  is  the  case. 
On  bringing  a  positively  charged  body  near  an  insulated 
conductor  there  is  a  separation  of  the  electricities  on  that 
conductor,  and  on  connecting  it  to  earth  positive  electricity 
passes  away,  hence  before  the  connexion  the  insulated  con- 
ductor had  a  positive  potential.  The  presence  of  the  positively 
charged  body  in  its  neighbourhood  immediately1  raised  its 
potential. 

"    18.     Importance  of  the  Insulating  Medium.    We 

have  seen  that  there  is  a  certain  analogy  between  a  number 
of  reservoirs  filled  with  compressed  air  and  a  number  of 
electrical  conductors,  but  the  analogy  does  not  carry  us  far. 
We  may  make  it  more  complete  thus. 

Imagine  a  lump  of  india-rubber  or  some  other  jelly-like 
elastic  substance  with  a  number  of  cavities  in  it.  Suppose 
that  each  of  these  cavities  has  a  pressure-gauge  attached  and  is 
filled  with  water.  Adjust  the  quantity  of  water  in  each  until 
the  gauges  all  read  alike.  The  cavities  represent  a  number  of 
conductors  and  the  india-rubber  the  insulating  medium  be- 
tween them.  Increase  the  pressure  in  one  of  the  cavities  by 
pumping  more  water  in  or  otherwise.  The  increased  pressure 
will  be  resisted  by  the  elasticity  of  the  medium,  and  the 
pressure  everywhere  throughout  the  medium  will  be  increased 
in  the  substance  of  the  elastic  medium  as  well  as  in  the  cavities ; 
the  gauges  will  all  read  differently,  the  increase  of  pressure  in 
any  cavity  will  depend  on  its  size  and  its  position  as  well  as 
on  the  elastic  properties  of  the  medium. 

In  electrical  language  one  of  the  conductors  has  been 
charged,  its  potential  has  been  raised ;  this  produces  a  change 
of  potential  everywhere ;  in  the  conductor  itself,  in  the 
surrounding  medium  or  dielectric,  and  in  neighbouring  con- 
ductors, this  change  depends  on  the  nature  of  the  medium 
as  well  as  on  the  size  and  position  of  the  conductors. 

Just  as  the  increase  of  pressure  in  a  cavity  in  the  india- 
rubber  sets  up  a  state  of  stress  throughout  the  mass  so  we 
may  look  upon  the  electrification  of  a  conductor  as  the  pro- 

1  It  is  probable  that  time  is  required  for  the  change  but  the  time  is 
extremely  short. 


17-19]       ELECTRICITY   AS   A    MEASURABLE   QUANTITY         31 

duction  of  a  state  of  stress  in  the  insulating  medium  round 
the  conductor ;  where  the  continuity  of  the  medium  is  broken 
by  the  presence  of  other  conductors  this  state  of  stress 
manifests  itself  in  the  phenomenon  we  call  the  electrification 
of  the  conductors.  From  this  point  of  view  to  produce  an 
electric  charge  at  any  point  is  to  throw  the  insulating  medium 
round  the  point  into-  a  state  of  stress,  and  the  electric 
attractions  and  repulsions  we  have  observed  are  the  con- 
sequences of  this  stress.  The  insulating  medium  and  not  the 
conductor  is  the  important  factor ;  it  has  the  power  of 
resisting  the  stress  and  possesses  what  we  may  call  electric 
rigidity ;  in  the  conductor  this  power  is  wanting,  the  stress 
breaks  down  as  soon  as  it  is  applied,  all  points  in  the  conductor 
are  at  the  same  electric  pressure  or  potential. 

This  is  the  more  modern  view  of  electricity  and  electric 
force,  and  in  it  the  importance  of  the  dielectric  is  clear. 

This  view  is  due  to  Faraday  and  was  developed  by  Clerk -Maxwell. 
We  might  carry  the  analogy  further  but  it  is  hardly  necessary  to  do  this 
at  present. 

19.  Equipotential  Surfaces.  A  positively  charged 
conductor  produces  electric  potential  at  all  points  in  its 
neighbourhood,  while  all  points  on  the  conductor  are  at  the 
same  potential.  The  surface  of  the  conductor  is  said  to  be  an 
equipotential  or  level  surface.  Consider  now  one  such  body, 


Fig.  13. 


32  ELECTRICITY  [CH.  II 

Figure  13,  the  potential  falls  off  as  we  recede  from  its  surface. 
Starting  outwards  from  the  body  along  any  line  I'1\P<,,  etc. 
we  can  find  a  series  of  points  J\,  /J2,  such  that  the  difference 
of  potential  between  any  two  consecutive  points  is  unity. 
Thus  if  the  potential  at  P  be  20,  the  potential  at  Pl  etc.  will 
be  19,  18,  17,  etc. 

We  can  do  the  same  for  all  points  Q,  R,  S,  etc.,  on  the 
surface  of  the  conductor  ;  we  thus  find  a  number  of  points, 
Qit&L»Si>  etc.  all  at  potential  19;  if  we  suppose  all  these  points 
joined  we  have  an  imaginary  surface  surrounding  the  body  at 
all  points,  of  which  the  potential  is  19;  this  is  called  an 
equipotential  surface. 

Similarly  by  joining  P2,  $2,  Rz,  etc.,  we  get  a  third  equi- 
potential surface  at  potential  18,  and  so  on;  these  surfaces 
surround  the  conductor  like  the  coats  of  an  onion. 

DEFINITION.  An  Equipotential  or  Level  Surface 
is  a  surface  at  all  points  of  which  the  potential  has  the  same 


We  have  described  the  drawing  of  an  equipotential  surface 
for  a  single  conductor;  if  we  have  a  number  of  conductors 
we  can  still  conceive  of  corresponding  equipotential  surfaces ; 
in  cases  in  which  it  is  possible  to  draw  these,  the  problem  of 
determining  the  distribution  of  electric  force  throughout  the 
field  can  be  solved. 

2O.  Lines  of  Force.  There  is  electric  force  at  any 
point  in  the  neighbourhood  of  a  charged  conductor  or  system  of 
charged  conductors,  and  at  each  point  this  force  has  its  own 
definite  direction. 

The  following  experiment  will  indicate  in  a  roughjnanner 
the  direction. 


Take  a  short  bit  of  cotton  thread  about  a  centimetre  long 
and  tie  a  long  thin  silk  fibre  to  its  centre.  Bring  the  cotton 
by  means  of  the  silk  fibre  near  a  charged  conductor,  it  will 
set  in  a  definite  position,  and  the  direction  of  its  length  is 
approximately  that  of  the ^  line  of  force-  at-its^  centre. 

For  the  cotton  is  a  conductor,  it  becomes  electrified  by 
induction  when  brought  near  the  charged  system,  becoming 


19-20]      ELECTRICITY  AS   A   MEASURABLE   QUANTITY        33 

positive  at  one  end,  negative  at  the  other;  the  positive  end  is 
pushed  in  one  direction  by  the  electric  force,  the  negative  end 
is  pulled  in  the  opposite  direction;  the  equilibrium  position 
will  be  found  when  the  lines  of  action  of  these  two  forces 
coincide,  that  is  when  the  length  of  the  bit  of  cotton  lies 
along  the  line  of  action  of  the  force.  In  this  manner  the 
fact  that  there  is  electric  force  at  all  points  of  the  field  can  be 
shewn,  and  its  direction  roughly  indicated. 

Now  let  AB,  Fig.  14,  be  one  position  of  the  test  thread. 
Suppose  the  thread  moved  until  A  comes  into  the  position  B 


Fig.  14. 

it  will  take  up  another  position  EC,  then  move  it  again  till 
the  end  originally  at  A  comes  to  C,  we  can  find  a  third 
position  CD  and  so  on.  Each  of  these  short  lines  AB,  £C, 
etc.,  indicates  the  direction  of  the  electric ^orce  at  its  centre; 
if  the  thread  A  B  is  very  small  these  short  bits  make  up  a 
f  itinuous  line,  usually  curved,  and  this  line  has  the  property 
that  its  direction  at  each  point  of  its  length  gives  the  direction 
of  the  resultant  electric  force  at  that  point.  Such  a  line  is 
called  a  line  of  force. 

DEFINITION.  A  Line  of  Force  in  an  electric  field  is  a 
line  such  that  its  direction  at  each  point  of  its  length  gives 
the  direction  of  the  resultant  electric  force  at  that  point. 

The  method  of  tracing  the  lines  of  force  just  described 
can  not  be  carried  out  satisfactorily  in  practice,  although  as 
we  shall  see,  under  Magnetism,  a  similar  method  is  quite 

G.  E.  3 


34  ELECTRICITY  [CH.  II 

practicable  and  of  great  value  for  tracing  lines  of  magnetic 
force ;  the  experiment,  however,  although  not  suited  to  give 
accurate  results,  is  sufficient  to  shew  the  presence  of  electric 
force  throughout  the  field  and  to  give  a  general  indication  of 
its  direction,  shewing  that  it  acts  along  lines  which  are  usually 
curved. 

^  21.  Lines  of  Force  and  Equipotential  Surfaces. 
If  we  trace  a  line  of  force  we  find  that  it  begins  and  ends  on  a 
charged  surface,  passing  in  all  cases  in  the  direction  in  which 
the  potential  falls,  so  that  if  the  potential  at  one  point  A  on  a 
line  of  force  is  above  that  at  another  point  B  the  direction  of 
the  line  is  from  A  to  B.  The  electrification  at  one  end  of  the 
line  will  be  found  to  be  positive,  that  at  the  other  end  of  the 
line  negative,  and  the  line  goes  from  the  positive  to  the  negative. 
Moreover,  it  will  be  clear  from  the  direction  in  which  the 
test  thread  sets  itself,  that  the  line  of  force  meets  the  sur- 
faces which  terminate  it  perpendicularly. 

This  leads  us  to  two  important  propositions : 

PROPOSITION  2.  The  two  extremities  of  a  line  of  force  rest 
on  oppositely  electrified  surfaces.  This  is  verified  by  direct 
experiment. 

If  we  connect  these  two  surfaces  by  a  conducting  wire  the 
extremities  of  the  line  of  force  may  be  supposed  to  move  up 
together  along  the  wire,  and  the  line  to  close  down  on  the  wire. 

PROPOSITION  3.     The  direction  of  a  line  of  force  is  perpen-\ 
dicular  to  that  of  all  the  equipotential  surfaces  which  it  meets. 

So  far  as  concerns  the  two  surfaces  on  which  <EFe~line"of 
force  ends,  and  which  are  of  course 
equipotential,  this  also  is  verified 
by  experiment.  A  theoretical  proof 
applying  to  all  equipotential  surfaces 
intersected  by  the  line  of  force  may 
be  given  thus  : 

Let-P,  Fig.  15,  be  any  point  on  an 
equipotential  surface,    and    Q    any   other          ^^^  J^ 

point  on  the  surface  adjacent  to  P.  Then 
since  P  and  Q  are  at  the  same  potential 
there  is  no  force  tending  to  move  electricity  Fig.  15. 

from  P  to  Q.  The  electrical  force  nt  P 
therefore  must  be  at  right  angles  to  PQ.  Hence  its  direction  is 


,TY 

20-22]      ELECTRICITY    AS   A    MEASURABLE   QUANTITY        35 

.     ^^**-^ 

perpendicular  to  the  surface,  in  other  words,  a  line  of  force  is  perpen- 
dicular to  any  equipotential  surface  which  it  meets. 

22.  Forms  of  Lines  of  Force.  The  student  will 
be  helped  in  this  view  of  electrical  action  if  he  can  picture 
to  himself  the  forms  of  the  lines  of  force  in  some  simple  cases. 
The  simplest  case  is  that  of  a  positively  charged  sphere  placed 
in  a  large  room,  the  lines  of  force  for  such  a  system,  Fig.  16, 


Fig.  16. 

are  straight  lines  radiating  outwards  from  the  centre  of  the 
sphere,  the  equipotential  surfaces  shewn  by  the  dotted  lines 
are  spheres  concentric  with  the  charged  sphere  ;  these  results 
follow  at  once  from  the  symmetry  of  the  system.  The  walls 
of  the  room  are  conductors  and  the  lines  of  force  end  on  them. 
Each  line  terminates  in  a  negative  charge  equal  to  the 
positive  charge  at  the  point  from  which  it  starts ;  the  total 

3—2 


36 


ELECTRICITY 


[CH.  II 


negative  charge  on  the  walls  is  equal  in  amount  to  the  positive 
charge  on  the  ball. 

Suppose  how  that  another  uncharged  and  insulated  con- 
ducting ball  is  brought  into  the  room ;  some  of  the  lines 
of  force  converge  on  to  this  ball,  see  Fig.  17,  meeting  it,  as 


Fig.  17. 

By  permission,  from  Watson's  Physics. 

shewn,  at  right  angles.  These  lines  end  in  a  negative  charge, 
so  that  the  side  of  the  ball  nearest  the  charged  conductor 
becomes  negatively  charged.  But  the  total  charge  on  the 
ball  is  zero,  hence  the  opposite  side  of  the  ball  acquires  a 
positive  charge  and  from  this  side  lines  of  force  equal  in 
number  to  those  entering  the  ball,  start  and  travel  towards 
the  walls  of  the  room.  The  dotted  lines  again  represent 
the  equipotential  surfaces. 


22]  ELECTRICITY   AS   A   MEASURABLE    QUANTITY  37 

In  Fig.  IS  is  shewn  the  result  after  the  second  ball  has 
been  connected  by  a  conducting  wire  to  the  walls. 

The  ball  has  thus  become  electrically  a  part  of  the  walls ; 
the  lines  of  force  between  the  ball  and  the  wall  have  their 
two  ends  on  the  same  conductor — this  is  impossible ;  the 


Fig.  18. 
By  permission,  from  Watson's  Physics. 

oppositely  electrified  ends  move  up  together ;  the  lines  close 
in  on  the  wire  and  collapse,  and  we  are  left  with  the  distribu- 
tion figured. 

The  tension  along  the  lines  of  force  in  both  these  cases 
tends  to  draw  the  two  conductors  together. 

The  distribution  on  two  similarly  charged  spheres  is  shewn 
in  Fig.  19. 

The  lines  run  from  the  spheres  to  the  wall ;   those  which 


ELECTRICITY 


[CH.  II 


start  from  either  sphere  towards  the  other  are  repelled  as  it 
were  by  the  corresponding  lines  from  the  second  sphere. 
The  point  P  between  the  spheres  on  the  line  joining  their 
centres  is  a  point  at  which  there  is  no  force  \  the  line  through 
this  point  perpendicular  to  the  axial  line  is  a  line  of  force. 


Fig.  19. 

The  tension  along  the  lines  between  the  spheres  and  the 
walls  draws  the  spheres  apart ;  they  apparently  repel  each 
other. 

Fig.  20  gives  two  oppositely  charged  spheres. 

^  A  number  of  the  lines  from  the  positive  sphere  bend 
round  and  converge  on  the  negative  sphere;  the  negative 
sphere  is  at  a  lower  potential  than  the  walls.  Hence  lines 


22-23]      ELECTRICITY   AS   A   MEASURABLE   QUANTITY        39 

pass  from  the  walls  to  it  while  other  lines  pass  from  the 
positive  sphere  to  the  walls.  The  tension  along  the  lines 
between  the  two  spheres  draws  them  together ;  they  attract 
each  other.  The  P  is  again  a  point  of  zero  force. 


Fig.  20. 

23.     The  Field   of  Force.     We   have   thus  another 
method  of  picturing  the  field  due  to  a  charged  conductor ;  we 


40  ELECTKICITY  [CH.  II 

ma}'  think  of  it  as  permeated  by  the  lines  of  force  due  to  that 
conductor,  lines,  that  is,  along  which  the  electric  force  due  to 
the  charge  on  the  conductor  acts.  If  we  start  outwards,  along 
one  of  these  lines,  and  follow  it  up,  we  shall  find  it  will  end 
somewhere  in  a  negative  charge  on  some  other  conductor; 
between  these  two  charges  there  is  apparently  a  force  of 
attraction.  If  we  wish  to  examine  further  the  mechanism  by 
which  the  force  of  attraction  is  produced  we  may  find  it  in  the 
insulating  medium  between  the  two.  We  have  already  seen 
that  the  electrification  of  a  conductor  may  be  the  manner  in 
which  we  recognise  a  state  of  stress  set  up  in  the  dielectric 
round  the  conductor.  Let  us  suppose  that  this  state  of  stress 
is  such  as  to  tend  to  cause  the  dielectric  to  contract  every- 
where along  the  lines  of  force  and  to  expand  everywhere  at 
right  angles  to  them.  Such  a  tendency,  if  of  the  proper 
amount,  would,  it  may  be  shewn,  produce  exactly  the  forces 
in  the  field  which  experiment  shews  to  exist. 

If  we  are  considering  the  force  between  two  small  charged 
bodies  we  may  either  speak  of  the  direct  attraction  between 
the  two,  or  we  may  picture  to  ourselves  the  lines  of  force 
which  join  them,  and  think  of  the  dielectric  as  tending  to 
shrink  up  along  these  lines  and  so  draw  the  two  bodies 
together ;  each  line  of  force  has  a  tendency  to  contract 
or  close  up  on  itself  like  a  stretched  elastic  thread.  Like  the 
thread  also,  it  tends  to  swell  laterally  at  the  same  time  as  it 
shrinks  longitudinally. 

v  This  method  of  considering  electrical  action  is  due  to 
Faraday ;  according  to  it  as  we  see  everything  depends  on  the 
properties  of  the  dielectric. 

24.  Mechanical  Illustrations.  We  may  illustrate 
the  matter  further  thus  : 

Suppose  we  have  two  balls  connected  by  a  number  of  fine 
elastic  threads  ;  we  may  imagine  the  threads  to  be  carried 
over  pullies,  or  otherwise  guided  in  some  way  so  that  they 
are  not  all  straight,  but  pass  from  the  one  body  to  the  other 
in  curved  lines. 

Let  the  bodies  be  pulled  slightly  apart  so  as  to  stretch 
these  threads.  Then  to  an  observer  who  was  not  conscious 


23-25]      ELECTRICITY    AS   A   MEASURABLE   QUANTITY       41 

of  the  threads  it  would  appear  that  there  was  an  attractive 
force  acting  between  the  bodies  across  the  empty  space  which 
separates  them  ;  one  who  could  recognise  the  existence  and 
functions  of  the  threads  would  see  that  the  attractive  force 
was  the  manifestation  of  the  stress  set  up  by  stretching  them. 
Or  again,  to  give  one  more  illustration,  we  might  look  upon 
a  line  of  force  as  a  string  of  little  cells  each  filled  with  liquid 
connecting  the  two  electrified  bodies  like  beads  upon  a  thread. 
Now  suppose  that  these  cells  can  be  made  to  spin  rapidly 
about  the  line  of  force  as  an  axis.  Each  cell  will  tend  to 
flatten  itself  along  the  axis  and  to  expand  in  directions  per- 
pendicular to  the  axis.  The  lines  of  force  will  tend  to  contract 
along  their  length  and  to  squeeze  each  other  outwards  in 
other  directions.  There  will  be  a  tension  in  the  dielectric  along 
the  lines  of  force,  and  a  pressure  perpendicular  to  them. 
The  stress  which  we  know  exists  in  an  electric  field  might  be 
produced  thus. 

25.  Theory  of  Potential  applied  to  Electrical 
Attraction.  In  the  earlier  Sections  we  have  already  described 
and  explained  various  experimental  results  on  the  hypothesis 
of  mutual  attractions  or  repulsions  between  electrified  bodies. 
Let  us  consider  how  some  of  these  are  modified  if  we  intro- 
duce the  notions  of  potential  and  lines  of  force.  Take  for 
example  the  phenomena  of  induction  when  a  positively  charged 
sphere  is  brought  near  an  insulated  conductor. 

We  know  that  (i)  the  sphere  produces  electrical  potential 
or  pressure  at  all  points  in  its  neighbourhood  and  that  this 
potential  falls  off  as  we  move  away  from  the  sphere1. 

(ii)  All  points  on  a  conductor  in  electrical  equilibrium 
are  at  the  same  potential. 

(iii)  Positive  electricity  passes  along  a  conductor  from 
points  at  high  to  points  at  low  potential. 

In  Fig.  21  let  the  consecutive  circles  indicate  the  equi- 
potential  lines  due  to  the  sphere  before  the  insulated  conductor 
is  brought  near,  and  let  the  numerals  10,  9,  8,  etc.  represent 

1  This  last  law  has  not  been  directly  proved  up  to  this  point,  but  see 
Section  26. 


ELECTRICITY 


[CH.  II 


the  potentials  of  these  surfaces.     On  bringing  the  conductoi 
near  it  cuts  a  number  of  these  surfaces ;    if  there  were  nc 


Fig.  21. 

change  in  potential  the  potential  of  one  point  on  the  con 
ductor  would  be  8,  of  another  7,  of  another  6,  and  so  on. 
This  however  is  impossible. 

Positive  electricity  passes  from  one  end  of  the  conductor, 
lowering  its  potential,  to  the  other  which  has  its  potential 


Fig.  22. 

raised;    the   result    is   that  the  first   end  is    left  negativel} 
charged,   while  the    second    acquires    positive   electrification 


25]  ELECTRICITY   AS   A   MEASURABLE   QUANTITY          43 

This  transference  goes  on  until  the  whole  conductor  is  reduced 
to  the  same  potential  and  the  equipotential  surfaces  due  to 
the  sphere  are  distorted  in  the  process  until  we  reach  the 
condition  represented  in  Fig.  22. 

Lines  of  force  now  pass  from  this  sphere  to  the  conductor 
and  the  tension  along  these  is  shewn  in  the  attraction  ob- 
served between  the  two. 

It  should  be  noticed  in  this  case  that  the  potential  of  the  conductor  is 
positive  though  it  is  negatively  charged  at  one  end. 

The  leaves  of  a  gold-leaf  electroscope  when  charged 
positively  are  at  a  higher  potential  than  the  walls  of  the 
case ;  these  if  coated  with  gauze  or  tinfoil  in  connexion  with 
the  table  are  at  zero  potential.  Lines  of  force  therefore  pass 
from  the  leaves  to  the  walls  and  the  tension  along  these  lines 
causes  the  leaves  to  stand  apart. 

If  we  suppose  the  electroscope  charged  by  induction  from 
a  positively  charged  body  we  may  represent  the  condition  of 
affairs  thus.  Lines  pass  from  the  positively  charged  body 
to  the  disc  or  knob  which  becomes  negatively  electrified ; 
a  positive  charge  is  thus  driven  to  the  leaves,  and  lines  of 
force  pass  from  them  to  the  cage.  The  tension  along  these 
lines  is  sufficient  to  draw  the  leaves  apart ;  hence  the  diver- 
gence which  is  observed. 


CHAPTER   III. 

MEASUREMENT   OF  ELECTRIC   FORCE  AND 
POTENTIAL. 

J  26.  Law  of  Force.  Up  to  the  present  it  has  not 
been  necessary  to  consider  what  is  the  force  between  two 
electrified  bodies  or  how  the  potential  in  the  neighbourhood  of 
a  charged  body  is  to  be  measured. 

We  have  learnt  that  two  bodies  charged  with  like  elec- 
tricities repel  each  other,  and  a  very  little  observation  is 
sufficient  to  shew  that  the  force  increases  as  the  distance 
between  the  bodies  is  decreased  ;  thus  the  divergence  of  the 
leaves  of  an  electroscope  caused  by  the  presence  of  a  charged 
body  increases  as  the  body  approaches  the  electroscope. 

We  must  now  proceed  to  consider  according  to  what  law 
the  force  decreases. 

In  the  first  place,  as  already  stated,  we  have  no  means 
of  observing  the  action  between  two  charges  of  electricity ; 
all  we  can  do  is  to  measure  the  force  between  two  bodies 
charged  with  electricity.  This  measurement  was  first  made 
by  Coulomb  by  aid  of  a  piece  of  apparatus  called  a  torsion 
balance1,  which  is  described  in  Section  59,  and  it  follows  as  a 
result  of  the  experiments  that  when  two  similarly  charged 
small  bodies  are  at  a  distance  apart  which  is  great  compared 

1  Among  the  many  interesting  pieces  of  historical  apparatus  in  the 
Paris   Exhibition   of   1900   was  the   original   torsion   balance   used   by 
•  Coulomb. 


26-27]  MEASUREMENT   OF   ELECTRIC    FORCE  45 

with  their  dimensions,  there  is  a  repulsion  between  them 
which  is  proportional  to  the  product  of  their  charges  and 
inversely  proportional  to  the  square  of  the  distance  between 
them.  Hence  Coulomb's  law  may  be  expressed  thus : 


'  LAW  OF  ELECTRIC  FORCE.  Consider  two  small 
charged  with  quantities  e,  e  of  electricity  and  placed  at  a 
distance  r  apart,  the  distance  r  being  great  compared  with  the 
size  of  either  body.  Then  there  is  a  force  of  repulsion  between 
these  bodies  which  is  proportional  to  ee/r2. 

Now  we  suppose  that  this  law  which  is  shewn  to  be  true 
within  the  limits  of  experimental  error  for  two  small  bodies 
would  be  accurately  true  for  two  electrified  points.  And  in 
the  mathematical  theory  of  electricity  we  treat  it  as  the  law 
of  force  between  the  particles  of  electricity,  and  our  real 
proof  of  the  law  lies  in  the  fact  that  the  results  of  complicated 
experiments  can  be  determined  by  calculation  on  the  as- 
sumption of  the  truth  of  this  law  and  are  found  to  agree  with 
observation  when  the  experiments  are  performed. 

'  In  particular  Faraday's  result  that  there  is  no  electric 
force  inside  a  hollow  closed  conductor  which  does  not  contain 
an  insulated  charged  conductor  is  consistent  with  this  law  of 
force,  and  with  this  law  only,  and  it  is  this  fact  which  forms 
the  real  basis  of  our  belief  in  the  law. 

v  27.   Effect  of  the  Dielectric;  Inductive  Capacity. 

We  have  said  already  that  according  to  the  modern  view  of 
electricity  the  action  between  electrified  bodies  depends  in 
great  measure  on  the  dielectric  or  insulating  medium  which 
separates  them,  and  it  is  found  that  the  force  between  two 
charged  bodies  can  be  changed  by  changing  this  medium 
without  altering  the  charge  or  position  of  either  body.  Ex- 
periments to  illustrate  this  are  described  in  Section  42. 

Suppose,  for  example,  that  it  is  possible  to  measure  the  force 
between  two  bodies  when  separated  by  air  and  that  without 
making  any  other  change  the  air  is  replaced  by  paraffin  oil. 
Then  it  will  be  found  that  the  force  is  about  half1  as  great  as 
it  was. 

1  To  be  accurate  it  is  reduced  in  the  ratio  1  to  1-92. 


46  ELECTRICITY  [CH.  Ill 

If  for  paraffin  oil  we  substitute  sulphur  then  the  force 
will  be  reduced  in  the  ratio  1  to  3-97,  it  will  be  about  one- 
fourth  of  what  it  was  in  air.  This  result  is  expressed  by 
introducing  into  the  law  of  force  a  factor  to  express  the  action 

the  dielectric.  This  factor  is  known  as  the  inductive 
capacity  of  the  dielectric.  If  we  denote  it  by  K  and  if  F  be 
the  force  between  two  particles  with  charges  e,  e  at  a  distance  r 
apart,  then  we  have  as  a  more  complete  statement  of  the  law 
the  result  that 


^  Since  in  a  very  large  number  of  experiments  air  is  the 
dielectric  medium,  it  is  found  convenient  to  assume  that  its 
inductive  capacity  is  unity,  and  to  measure  the  inductive 
capacities  of  other  media  in  terms  of  this.  This  is  known  as 
the  electrostatic  system  of  measurement. 

•  The  ratio  of  the  inductive  capacity  of  a  medium  to  the 
inductive  capacity  of  air  is  known  as  the  Specific  Inductive 
Capacity  of  the  medium. 

v  On  the  electrostatic  system,  since  the  inductive  capacity  of 
air  is  assumed  to  be  unity,  it  follows,  that  the  numerical 
measures  of  the  inductive  capacity  of  a  medium  and  of  its 
specific  inductive  capacity  are  the  same. 

In  considering  this  subject  the  student  should  compare  the  definitions 
of  density  and  specific  gravity;  when  water  is  the  standard  the  numerical 
measures  of  these  two  quantities  are  the  same. 

28.  Units  of  Measurement.  We  have  not  up  to 
the  present  defined  in  any  way  the  units  in  which  the 
quantities  are  measured.  As  usual  we  employ  the  c.  G.  s. 
system  ;  the  unit  of  distance  is  the  centimetre,  the  unit  of 
force  the  dyne;  the  unit  quantity  of  electricity  requires 
definition. 

We  have  the  law  that,  if  the  force  between  two  quantities 
e,  e  placed  in  air  at  a  distance  r  centimetres  apart  is  F  dynes, 
then 

F  =  —  -  dynes. 

.2        J 


27-29]  MEASUREMENT   OF   ELECTRIC    FORCE  47 

Now  let  the  quantities  e,  e  be  equal  and  let  them  be  such 
that  when  placed  at  a  distance  of  one  centimetre  apart,  the 
force  of  repulsion  is  1  dyne. 

Then  in  the  above  expression  we  have  e  —  e',  F—\  dyne, 
r  =  1  cm. 

e2 
Hence  1  =  —  . 

r 


Thus  in  this  case  e  must  be  the  unit  quantity  of  electricity. 
Or  in  other  words, 


The  Unit  Quantity  of  Electricity  is  that 
quantity  which  placed  at  a  distance  of  1  centimetre  in  air  from 
an  equal  and  similar  quantity  repels  it  with  a  force  of  1  dyne. 

On  this  assumption  then  as  to  the  unit  quantity  we  h? 
the  law  that  on  the  electrostatic  system  in  air 

F=  ~T  dynes; 

and  in  a  medium  of  specific  inductive  capacity  A', 

r,      I  ee' 

F=  —  -j  dynes. 

Thus  if  we  call  Fl  the  force  between  the  two  quantities 
e,  e'  when  at  a  distance  r  centimetres  apart  in  air,  and  F2  the 
force  between  the  same  two  quantities  when  in  a  medium  of 
specific  inductive  capacity  K, 

then  clearly  Fl  -  ee'jr\      F2  =  ee'/Kr2. 

Hence  Fl  =  KF2  or  K=  F,/F2. 

Thus  the  specific  inductive  capacity  of  a  dielectric  is  the 
ratio  of  the  force  between  two  charges  in  air,  to  the  force 
between  the  same  two  charges  when  in  the  dielectric  when 
the  distance  between  the  two  charges  remains  unaltered,  which 
explains  the  definition  given  above. 

29.  Resultant  Electric  Force  at  a  point;  Electric 
Intensity.  At  any  point  in  the  neighbourhood  of  a  charged 
conductor  there  is  electrical  force ;  if  we  place  at  that  point  a 


48  ELECTRICITY  [CH.  Ill 

small  body  charged  with  a  quantity  e  of  electricity,  then  this 
charge  is  repelled  or  attracted  by  the  charge  on  the  conductor, 
and  we  can  in  theory  at  least  calculate  the  amount  of  the 
force  by  calculating  the  force  exerted  by  each  element  of  the 
charge  on  the  conductor,  and  finding  the  resultant  of  these. 
This  resultant  force  will  be  proportional  to  the  charge  on  the 
small  conductor,  for  each  component  force  is  so ;  and  we  may 
denote  it  by  lie'.  Thus  R  is  the  force  which  would  be  exerted 
on  a  small  body  at  the  point  in  question  when  carrying  a 
unit  positive  charge. 

The  quantity  R  is  known  either  as  the  resultant  electric 
force  or  as  the  electric  intensity  at  the  point. 

It  must  be  noticed  however  that  if  we  were  actually  to  bring  a  small 
body  so  charged  near  to  the  conductor,  the  distribution  of  electrification 
would  be  disturbed ;  in  calculating  the  force  theoretically  we  suppose 
this  disturbance  not  to  occur,  so  that  the  resultant  force  could  not  really 
be  measured  experimentally  by  bringing  up  a  small  charged  body  and 
determining  the  force  acting  on  it. 

We  are  thus  led  to  the  following : 

DEFINITION.  The  Resultant  Electric  Force  or  tKe 
Electric  Intensity  at  a  point  is  the  force  acting  on  a 
small  body  charged  with  a  unit  positive  charge  when  placed  at 
the  point,  the  electrification  of  the  rest  of  the  system  being 
supposed  undisturbed  by  the  presence  of  this  unit  charge. 

If  the  electrical  field  be-due  to  A  charge  ^^dhce~nTrated  at 
a  distance  r  from  the  point  at  which  the  resultant  force  is  to 
be  measured,  we  know  that  the  foi*ce  between  two  small  bodies 
carrying  charges  e  and  e  respectively  is  ee'/rz.  The  electric 
intensity  is  the  force  on  a  small  body  carrying  unit  charge. 
Hence  we  see,  by  putting  e  equal  to  unity,  that  in  this  case 
the  electric  intensity  is  e/r*. 

Thus  the  Resultant  Electric  Force  or  Electric 
Intensity  at  any  point  due  to  a  charge  e  concentrated  at 
a  distance  of  r  centimetres  from  that  point  is  e/r2. 

As  an  illustration  of  the  magnitude  of  the  electrostatic 
unit  we  may  note,  that  if  two  equal  small  conducting  spheres 
each  \  gramme  in  weight  are  suspended  from  the  same  point 
by  two  silk  fibres  of  inappreciable  weight  100  centimetres  in 
length  and  equally  electrified  until  they  separate  to  a  distance 


29-30]  MEASUREMENT   OF   ELECTRIC    FORCE  49 

of   10  centimetres,  then  the  charge  on  each  must  be  about 
22  units. 

Examples.     (1)   Calculate  the  force  between  tivo  small  spheres  charged 
with  10  and  15  units  respectively  placed  at  a  distance  of  50  cm.  apart. 

10x15  .  3    . 

-^-  dyne  =  _  dyne. 

(2)  The  force  betiveen  the  two  spheres  in  Example  (1)  is  known  to 
be  '02  dyne.     What  is  the  specific  inductive  capacity  of  the  medium? 

The  specific  inductive  capacity  is  equal  to  the  ratio  of  the  force  in  air 
to  the  force  in  the  medium.     The  force  in  air  =  3/50  =  -06  dyne, 

»n/» 
.'.  specific  inductive  capacity  = —  =  3. 

(3)  How  far  apart  must  two  small  spheres  each  charged  ivith  100  units 
be  placed  in  air  in  order  to  repel  each  other  with  a  force  equal  to  the 
weight  of  one  gramme  ? 

The  weight  of  one  gramme  =  981  dynes.  Let  the  distance  be  r  centi- 
metres. 

Then  gs^100*100. 

r2 

.'.  r=100/\/98l=  100/31-32  =  3-19  cm. 

30.  Measure  of  Potential.  In  Section  15  we  have 
explained  in  a  general  manner  the  meaning  of  the  term 
Electrical  Potential,  but  the  account  there  given  will  not 
enable  us  to  measure  potential.  Now  we  have  seen  already 
that  there  is  a  close  analogy  between  the  relation  of  the  flow 
of  heat  and  of  difference  of  temperature  on  the  one  hand,  and 
that  of  electrical  force  and  difference  of  potential  on  the  other. 
Let  us  carry  this  rather  further. 

We  have  seen  (Glazebrook,  Heat,  §  141)  that  if  t  and  t'  are 
the  temperatures  of  two  points  d  cm.  apart,  and  if  the 
temperature  gradient  be  uniform  between  these  points,  then 
the  flow  of  heat  anywhere  between  the  points  is  constant, 
and  we  have  the  result  that  flow  of  heat  is  proportional  to 
(t  -  t')/d. 

Take  now  a  case  in  which  the  potentials  at  two  points 
d  centimetres  apart  are  V  and  V,  and  the  potential  gradient 
is  uniform.  Then  our  analogy  would  lead  to  this  result  that 
the  electrical  force  anywhere  between  the  points  is  constant 
and  is  proportional  to  (V  —  V')/d. 

G.  E.  4 


50  ELECTRICITY  [CH.  Ill 

Let  us  consider  the  result  of  assuming  this  law  to  hold, 
and  suppose  that  the  electrical  force  between  the  points  is  ft. 
Then  it  follows  that  ft  is  proportional  to  (V  —  V'r)/dt  and  if  we 
choose  our  units  of  measurement  aright  we  can  put 


Hence  V-V'  =  ftd. 

Now  ft  is  the  force  on  a  small  body  carrying  a  unit 
positive  charge,  and  ftd  is  the  work  done  in  moving  the  body 
a  distance  d  against  the  force. 

Thus  we  shall  obtain  a  result  quite  consistent  with  what 
precedes  if-we-adopt  the  following^  _  __  __ 

DEFINITION.  The  Difference  of  Potential  between  two 
points  A  and  B  is  the  work  done  in  moving  a  small  body 
carrying  a  unit  positive  charge,  from  B  to  A,  against  the 
electrical  force,  the  electrification  of  the  rest  of  the  system  being 
supposed  undisturbed  by  the  presence  jy£ijLhi£.  unit  charge. 

~~We~"have  been  leoT^Eo~ttris^dennition  by  considering  the 
case  of  a  uniform  force  and  the  analogy  with  the  flow  of  heat  ; 
by  adopting  it  in  general  we  are  enabled  to  measure  in  theory 
at  least  the  electrical  potential  at  any  point  of  the  electric 
field  due  to  a  charged  system.  Practical  methods  of  measuring 
potential  will  be  given  later. 

It  should  be  noticed  that  the  definition  gives  us  a  means 
of  measuring  difference  of  potential,  it  does  not  enable  us  to 
say  without  further  explanation  what  the  potential  at  a  point 
is  ;  if  however  we  go  sufficiently  far  from  our  charged  system 
the  force  due  to  it  will  be  very  small,  so  small  that  we  may 
neglect  it,  no  work  will  be  done  in  moving  our  unit  charge  so 
long  as  we  keep  far  enough  away  ;  all  points  therefore  at  a 
sufficient  distance  from  our  charged  system  are  at  the  same 
potential,  and  we  may  take  this  as  the  zero  of  potential  for 
these  purposes;  these  points  are  said  to  be  beyond  the 
boundaries  of  the  electrical  field  of  the  charged  system,  and 
in  this  case  the  potential  at  the  point  might  be  defined  as  the 
work  necessary  to  bring  a  small  body  carrying  a  unit  positive 
charge  from  beyond  the  boundaries  of  the  field  up  to  the 
point. 


30-32]  MEASUREMENT   OF   ELECTRIC   FORCE  51 

For  some  other  purposes  again  it  is  convenient  to  treat 
the  potential  of  the  earth  as  the  zero  of  potential ;  as  we  have 
said  already,  in  consequence  of  the  size  of  the  earth,  we 
cannot  alter  its  potential  by  any  finite  charge  which  we  can 
give  it ;  in  all  cases  in  which  we  speak  of  the  potential  of  a 
body  as  being  V  we  mean  that  it  is  V  units  of  potential  above 
some  body  taken  as  the  zero,  and  that  V  units  of  work  are 
necessary  to  move  a  small  body  carrying  unit  positive  charge 
from  that  zero  of  potential  to  the  body  in  question,  the 
electrification  of  the  rest  of  the  system  being  supposed  un- 
affected by  the  motion. 

31.  Unit    of   Potential.      It    is    clear    that    since 
potential    is    measured    by   work    done,    there   will    be   unit 
difference  of  potential  between  two  points  when  the  work 
done  in  moving  ulfiit  charge  is  the  unit  of  work,  the  unit  of 
work  is  called  as  we  know1  the  erg,  and  is  the  work  done  on 
a  particle  which  is  moved  one  centimetre  against  a  force  of 
one  dyne. 

DEFINITION.     There  is  Unit  Difference  of  Potential 

between  two  points  when  one  erg  of  work  is  done  in  moving  a 
small  body  carrying  unit  positive  charge  from  one  point  to  the 
other,  the  electrification  of  the  rest  of  the  system  being  supposed 
unaffected  by  the  motion. 

In  dealing  with  the  subject  of  potential  we  might  have  commenced 
with  the  Definition  given  in  Section  30,  thus  introducing  the  potential 
at  a  point  as  a  quantity  which  we  can  measure,  and  which  has  an 
important  place  in  electrical  theory.  Such  a  course  would  have  tended 
to  greater  precision.  Since  however  the  general  idea  of  potential  can  be 
utilized  as  has  been  done  in  Section  25,  without  a  strict  definition  of  its 
measure,  the  course  adopted  has  seemed  the  better  in  the  hope  that  some 
students  who  have  a  difficulty  in  grasping  the  idea  of  work  done  may  yet 
not  be  prevented  from  using  this  conception  of  potential  as  a  quality  of 
a  body  on  which  the  direction  of  the  electric  flow  depends. 

32.  Calculation  of  Potential.     As  we  have  already 
said  the  difference  of  potential  between  any  two  points  can  be 
calculated  in  theory  when  we  know  the  field,  let  us  exemplify 
this,  calculating  the  potential  at  a  point  P,  Figure  23,  due  to 
a  charge  Q  at  a  point  0. 

1  Dynamics,  Section  124. 

4—2 


52  ELECTRICITY  [CH.  Ill 

Join  OP  and  let  OP  equal  r. 

Produce  OP  to  a  great  distance — beyond  the  limits  of  the 


Fig.  23. 

field  —  and  let  Plt  P2  be  two  points  on  OP  at  distances  j\  and  7*2 
respectively  from  0.  Then  the  force  at  Pl  is  Q/r*t  that  at  P2 
is  Q/r*,  and  if  P,  is  very  close  to  P2,  r*  is  very  nearly  equal  to 
rs2,  each  may  be  put  equal  to  r^,  thus  ultimately  we  may 
treat  these  forces  as  equal,  and  equal  to  $/Vjr2. 

Thus  when  the  distance  P^P*  which  is  equal  to  r2  —  r^ 
is  sufficiently  small,  the  work  done  in  bringing  unit  positive 
charge  from  P2  to  Plt  being  equal  to  the  product  of  the  force 
and  the  displacement,  is  Q  (r2  —  r-^jr-^r^  and  this  work  is  the 
difference  of  potential  between  Pl  and  P3. 

Hence  r^F,=  «^^>=  «-  «. 

7^2          /-!     r2 

By  taking  another  point  P3  beyond  P2,  but  very  near  to 
it,  we  can  get 


Thus  if  we  go  to  a  point  Pn  at  a  distance  rn  from  0  we 
have  the  equations 


32-34]  MEASUREMENT   OF   ELECTRIC   FORCE  53 

Hence  by  adding  these  we  have 


But  Pl  and  Pn  are  any  two  points  on  the  line  OP. 
Denoting  them  by  P  and  P  and  their  potentials  by  V  and  V 
we  have 


If  we  suppose  P  to  be  at  a  very  long  distance  away  we 
may  put  V  —  0,  and  since  /  is  infinite  Q/r'  is  zero.  Thus  we 
find 


We  have  thus  obtained  an  expression  for  the  potential  due 
to  a  charge  Q  concentrated  at  a  point. 

We  may  notice  that  the  potential  due  to  such  a  charge  at 
any  point  is  inversely  proportional  to  the  distance  of  the  point 
from  the  charge,  we  have  already  seen  that  the  resultant 
electrical  force  or  the  electric  intensity  is  inversely  pro- 
portional to  the  square  of  the  distance. 

33.  Equipotential  Surface.  We  may  use  the  above 
result  to  draw  the  equipotential  surfaces  and  lines  of  force 
due  to  such  a  charge. 

Thus,  since  the  value  of  the  potential  at  a  distance  r  is  Q/r, 
for  all  points  at  which  the  potential  has  a  definite  constant 
value  r  must  be  the  same  ;  such  points  therefore  are  at  the 
same  distance  from  the  charge,  that  is  they  lie  on  a  sphere. 
Hence  the  surfaces  of  equal  potential  are  spheres  ;  the  lines 
of  force  are  the  radii  of  the  spheres  as  shewn  in  Figure  16. 

Or  again  take  the  case  of  an  electrified  conducting  plane 
of  considerable  area.  The  plane  itself  is  equipotential,  and  the 
equipotential  surfaces  except  near  the  edges  will  be  parallel 
planes  while  the  lines  of  force  will  be  straight  lines  at  right 
angles  to  the  planes. 

34.    Potential  of  a  Charged  Conductor.   We  have 
already  defined  the  potential  at  a  point  as  the  work  done  in 


54  ELECTRICITY  [CH.  Ill 

bringing  a  small  body  carrying  unit  positive  charge  up  to  the 
point ;  now  the  point  may  be.  on  a  conductor,  and  if  this  is  so 
the  potential  at  the  point  will  be  the  potential  of  'the  conductor. 
Work  has  to  be  done  to  bring  the  unit  positive  charge  up  to 
any  point  on  the  conductor.  This  work,  if  we  suppose  the 
electrification  of  the  conductor  unaffected  by  the  presence  of 
the  unit  charge,  will  be  the  same  whatever  point  of  the 
conductor  be  approached,  and  the  amount  of  work  done 
measures  the  potential  of  the  conductor. 

35.  Relation    of   Charge   and   Potential.      The 

charge  on  a  conductor  will  be  connected  with  its  potential ; 
the  potential  of  a  conductor,  however,  as  we  have  seen  does 
not  depend  solely  on  its  charge,  but  also  on  the  condition  of 
the  other  conductors,  if  there  be  any,  in  its  neighbourhood  : 
let  us  suppose  that  aU  other  conductors  near  are  maintained  at 
zero  potential  by  being  connected  to  the  earth,  then  there  is  a 
simpje  relation  between  the  charge  and  potential  of  the  one 
insulated  conductor.  For  consider  such  a  charged  body ;  the 
force  at  any  point  in  its  field  is,  as  we  have  seen,  the  resultant 
of  the  forces  arising  from  its  electrical  charge  together  with  the 
forces  which  arise  from  the  induced  charges  on  the  neighbouring 
uninsulated  bodies. 

If  the  charge  at  each  point  of  the  surface  of  the  insulated 
conductor  be  doubled,  each  of  these  induced  charges  will  be 
doubled,  and  hence  each  of  the  various  component  forces  will 
be  doubled.  Thus  the  resultant  force  at  each  point  of  the 
field  is  doubled ;  hence  the  work  done  in  bringing  up  a  unit 
charge  to  that  point  will  be  doubled,  that  is,  the  potential  at 
each  point  of  the  field  is  doubled. 

"*  -     Hence  by  doubling  the  charge  on  the  insulated  conductor 
we  have  doubled  the  potential  everywhere. 

•l  Thus  if  all  other  conductors  in  its  field  are  at  -  zero 
potential,  the  potential  on  any  one  insulated  conductor  is  pro- 
portional to  its  charge. 

36.  Capacity  of  a  Conductor. 

DEFINITION.  The  ratio  of  ilie~ 'charge  on  an  insulated  con- 
ductor to  the  potential  of  that  conductor,  when  all  neighbouring 


34-37]  MEASUREMENT   OF   ELECTRIC    FORCE  55 

conductors  are  at  zero  potential,  is  found  to  be  constant  and  is 
called  the  Capacity  of  the  conductor. 

If  G  be  the  capacity  of  a  conductor,  Q  its  charge,  and  V  its 
potential,  other  conductors  being  at  zero  potential,  then  C  is  a 
constant  for  the  conductor  and  we  have  the  relation 


=  v> 

or  as  we  may  put  it 

Q=cr, 

J      If   the  neighbouring   conductors  be    not   earth-connected 
then  this  relation  does  not  hold. 

Suppose  now  that  we  charge  an  insulated  conductor  until 
it  acquires  unit  potential.  Then  in  the  above  equation  V  is 
unity,  and  Q  is  the  charge  required  to  raise  the  potential  of 
the  conductor  to  unity.  But  from  the  above  equation  since 
V  =  1  we  have 

C=Q. 

Hence  we  obtain  another  definition  of  the  electric  capacity 
of  a  conductor. 

DEFINITION.  The  Capacity  of  a  conductor  is  the  electrical 
I  charge  required  to  raise  the  conductor  —  all  other  conductors  in 
u  the  Jield  being  earth-connected  —  to  unit  potential.  '  *, 

i  "  We  may  compare  this  with  the  definition  of  capacity  for  heat1.  The 
capacity  for  heat  of  a  body  is  the  quantity  of  heat  required  to  raise  the 
temperature  of  the  body  by  unity  ;  we  again  have  an  analogy  between 
heat  and  electricity. 

37.  Energy  of  a  Charged  Conductor.  If  a  unit 
positive  charge  be  brought  up  to  a  conductor  at  potential  F, 
and  it  be  supposed  that  the  electrical  distribution  is  not  thereby 
altered,  the  work  done  is  by  definition  V  \  if  a  charge  q  be 
brought  up,  q  being  very  small,  so  that  its  presence  does  not 
disturb  the  electrical  distribution,  the  work  done  is  Vq. 

Now  suppose  we  have  an  uncharged  insulated  conductor, 
and  that  we  charge  it  by  bringing  up  a  series  of  equal  small 
charges  qlt  q2,  q3,  etc.  Let  its  potential  after  it  has  received 
these  successive  charges  become  V1}  V.2)  V9,  etc.  In  reality  as  the 

^  Glazebrook,  Heat,  p.  34. 


56 


ELECTRICITY 


[CH.  Ill 


charging  proceeds  the  potential  rises  gradually  from  Vl  to  F2, 
F2  to  F3  and  so  on.  Let  us  however  proceed  to  calculate  the 
work  on  the  assumption  that  the  change  is  sudden,  and  that 
during  the  interval  in  which  the  charge  <?2  for  example  is 
being  brought  up  the  potential  is  F2  and  so  on.  Then  the 
work  done  is  clearly 

s.r.+^r. +  ?.>,  +  ... 

This  can  be  represented  graphically  as  in  Fig.  24.     For  let 


R* 


Ra 


N 


N3      N4     N 


Fig.  24. 

a  distance  ON  measured  along  a  horizontal  line  represent 
the  charge  Q,  and  let  PN  measured  perpendicularly  to  ON 
represent  the  potential  V.  Divide  ON  into  a  large  number  of 
equal  parts  in  N19  Nz  etc.,  then  these  parts  represent  the 
small  charges  ql9  q2  etc.,  and  draw  -#",/*,,  NZP2  to  represent 
the  potentials  F,,  F2  after  the  respective  charges  have 
been  given  to  the  conductor.  Complete  the  parallelograms 
ON1P1R1,  N^NZPZRZ  as  shewn  in  Fig.  24. 

Then  area  ON^  Pl  ^  =  Fj  ql 

etc. 


Hence  the  work  done  in  charging  which  has  been  shewn 
to  be  equal  to 


37]  MEASUREMENT   OF   ELECTRIC    FORCE  57 

is  equal  to  the  sum  of  the  areas 


Now  if  the  distances  0^,  N-^N^  etc.  are  sufficiently  small 
the  points  Plt  P2...etc.  will  lie  on  a  continuous  line  OP1P2...P1 
and  the  sum  of  the  rectangular  areas  N^N^P^R^  etc.  will 
coincide  with  the  area  bounded  by  this  line,  and  the  lines  ON 
and  PN.  This  area  then  represents  the  work  done  in  charging 
the  conductor. 

But  since  the  potential  is  proportional  to  the  charge  it 
follows  that  OP1P<2...P  is  a  straight  line.  The  area  then 
representing  the  work  is  therefore  a  triangle  on  the  base 
ON  and  of  altitude  PN. 

The  area  of  this  triangle  is  \  PN  .  ON  or  J  VQ,  where  Q  is 
the  sum  of  the  small  quantities  qlt  q2,  etc. 

Hence  the  work  done  in  charging  a  conductor  to  potential 
V  with  a  charge  Q,  all  other  conductors  in  the  field  being  at 
zero  potential,  is  J  V.  Q. 

/  The  charged  conductor  has  acquired  energy,  and  its  energy 
or  capacity  for  doing  work  is  measured  by  the  work  done 
in  charging  it. 

Hence  the  energy  of  this  charged  conductor  is  J  V.  Q. 
Since  we  have  Q  =  CV  we  can  express  this  energy  E  in 
terms  either  of  Q  or  V  and  the  capacity. 
For  substituting  for  Q  we  obtain 


while  substituting  for  V  we  get 


Thus  the  work  done  in  charging  a  conductor  with  a  given 
charge  is  inversely  proportional  to  its  capacity,  while  the  work 
done  in  charging  it  to  a  given  potential  is  directly  proportional 
to  its  capacity. 

The  method  of  proof  should  be  compared  with  that  given  in 
Glazebrook,  Dynamics,  §  39,  for  finding  the  space  passed  over  by  a 
particle  moving  with  known  velocity,  and  the  formulae  obtained  with 
those  found  for  the  space  traversed  when  the  acceleration  is  constant, 


CHAPTER   IV. 

CONDENSERS. 

'    38.     Condensers.     The  term  condenser  is  used  for  a 
conductor   or   conductors    arranged    to   have    specially    large 
Capacity. 

To  explain  the  action  of  a  condenser  let  us  consider  the 
following  experiment. 

In  Fig.  25  A  and  B  represent  two  parallel  vertical  metal 


Fig.  25. 

plates  each  insulated ;  the  plate  A  is  connected  to  an  electro- 
scope, the  plate  B  is  moveable, — its  support  rests  in  a  groove, 
the  direction  of  which  is  perpendicular  to  the  planes  of  the 
two  plates, — and  it  is  placed  at  some  distance  from  A.  Such 
an  instrument  is  called  a  condenser. 

Electrify  A;  the  gold  leaves  diverge,  then  bring  B  near 
to  A,  the  divergence  of  the  leaves  decreases.  If  B  be  con- 
nected to  an  electroscope  when  at  a  distance  from  A,  the 
leaves  of  this  electroscope  will  diverge  as  B  is  brought  near 
to  A.  Moreover  it  will  be  found  that  the  sign  of  the  electri- 
fication of  both  electroscopes  is  the  same ;  this  can  be  shewn 
by  bringing  an  electrified  ebonite  rod  near  to  each  in  turn. 


38-39]  CONDENSERS  59 

Now  connect  B,  when  it  is  near  A,  to  earth,  the  leaves  of 
the  A  electroscope  will  collapse  still  further,  those  of  the 
B  electroscope  will  collapse  entirely. 

We  c*an  put  the  explanation  of  these  observations  in  various 
ways.  Thus,  suppose  A  is  electrified  positively  when  B  is  brought 
near,  negative  electrification  is  attracted  to  its  surface  near  A, 
and  positive  repelled  into  the  gold  leaves ;  the  negative  electricity 
attracts  the  positive  charge  of  A.  Part  of  the  charge  of  the 
gold  leaves  connected  to  A  is  thus  withdrawn,  and  they  diverge 
less  than  originally.  When  B  is  now  connected  to  earth,  the 
positive  charge  on  the  back  of  B  and  on  the  gold  leaves  of  the 
B  electroscope  is  repelled  to  earth.  More  of  the  positive  charge 
of  A  is,  in  consequence,  attracted  to  its  face,  the  charge  on 
the  gold  leaves  is  reduced,  and  they  collapse  still  further.  It 
is  as  though  the  presence  of  the  plate  B,  specially  if  it  be 
earth -connected,  condensed  the  positive  charge  of  A  on  to  the 
surface  opposite  B.  Hence  the  name  condenser.  The  charges 
on  the  opposing  faces  of  the  two  condenser  plates  are  sometimes 
spoken  of  as  bound-charges. 

We  can  put  the  matter  more  concisely  by  introducing  the 
idea  of  potential. 

39.     Explanation  of  the  Action  of  a  Condenser. 

When  the  plate  A  is  positively  electrified,  it  and  the  gold 
leaves  of  the  A  electroscope  are  at  a  definite  positive  potential, 
the  gold  leaves  therefore  diverge ;  the  plate  B  is  at  a  lower 
potential,  if  it  be  earth-connected  it  is  at  potential  zero. 
Suppose  in  the  first  case  that  B  is  insulated.  As  B  is  brought 
near  to  A  its  own  potential  rises,  but  the  presence  of  B,  a 
body  at  lower  potential  than  A,  reduces  the  potential  of  A, 
thus  the  gold  leaves  of  the  A  electroscope  become  less  divergent, 
while  those  of  the  B  electroscope  diverge,  being  raised  to  a 
positive  potential  by  the  presence  of  A. 

If  B  be  earth-connected  its  potential  is  zero,  its  presence 
therefore  reduces  the  potential  of  A  still  further,  and  the 
divergence  of  the  leaves  of  the  A  electroscope  is  still  more 
reduced. 

Thus  in  the  presence  of  B  an  additional  charge  is  required 
to  raise  the  potential  of  A  to  what  it  was  previously.  The 
capacity  of  A  is  raised  by  the  presence  of  B. 


60  ELECTRICITY  [CH.  IV 

40.  Capacity  of  a  Condenser.  We  have  already 
defined  the  capacity  of  a  conductor  as  being  the  ratio  of  its 
charge  to  its  potential,  when  all  neighbouring  conductors  are 
at  potential  zero.  When  B  is  at  some  distance,  the  composite 
conductor  composed  of  A  and  the  electroscope  has  a  certain 
capacity ;  on  bringing  the  earth  plate  B  near  the  total  charge 
on  this  conductor — A  and  the  electroscope — is  not  altered, 
but  its  potential  is  reduced,  hence  the  capacity  of  the  con- 
ductor is  increased,  and  this  increase  in  capacity  may  be  very 
large. 

If  instead  of  using  an  electroscope  to  do  the  experiment  with  we 
employed  a  quadrant  electrometer,  see  Section  62,  or  some  other 
instrument  for  the  measurement  of  potential,  we  might  compare  the 
potential  in  the  two  cases,  (i)  when  B  is  at  a  distance,  (ii)  when  it  is 
near  A,  and  from  this  comparison,  if  we  could  make  sure  that  there  had 
been  no  leak,  we  could  determine  the  ratio  of  the  capacities. 

A  slightly  different  definition  of  capacity  from  that  already 
given,  which,  however,  as  we  shall  see  shortly  is  quite  consistent 
with  it,  will  be  found  of  use  when  dealing  with  a  condenser. 
This  we  proceed  to  consider. 

In  a  case  in  which  all  the  lines  of  force  from  a  body  at 
potential  Fx  pass  to  another  body  at  potential  F2  it  is  found 


Fig.  26. 

as  the  result  both  of  theory  and  experiment  that  the  charge 
on  the  first  body  is  proportional  to  the  difference  of  potential 
between  the  two. 

The  case  is  only  fully  realised  when  a  body  A  (Fig.  26) 
containing  a  charge  Ql  at  potential  FT  is  completely  surrounded 


40]  CONDENSERS  61 

by  a  second  body  B  at  potential  F2,  when  this  happens  it  is 
found  that  the  ratio  Q^(  V1  —  F2)  is  constant.  Such  an  arrange- 
ment constitutes  a  condenser,  the  presence  of  the  outer  body  B 
increases  the  charge  requisite  to  raise  A  to  a  given  potential. 
The  constant  ratio  of  the  charge  to  the  difference  of  potential 
in  such  a  case  is  known  as  the  capacity  of  the  condenser. 
Hence  if  C  be  the  capacity  we  have 

ft      _.r 

y  _  y-'^y 

or  Q1  =  C(V1-VZ). 

The  fact  that  in  a  case  of  this  kind  we  double  the  difference 
of  potential  between  A  and  B  by  doubling  Ql  may  be  seen 
thus.  Let  us  suppose  that  B  completely  surrounds  A  and 
that  it  has  a  charge  Q2. 

Then  since  the  charge  on  B  produces  no  force  at  any  point  in 
the  space  within  the  inner  surface  of  7?,  the  force  at  any  point 
in  this  space  is  entirely  due  to  the  charge  on  A.  If  we  double 
the  surface  density  at  each  point  of  A,  and  thus  double  its 
charge,  we  double  the  force  everywhere  between  A  and  B. 
Thus  the  work  necessary  to  move  a  unit  charge  from  B  to  A 
is  doubled.  But  this  work  is  Fj  —  F2.  Hence  by  doubling 
the  charge  on  A  we  double  the  difference  of  potential  between 
A  and  JB,  or  the  ratio  of  the  charge  on  A  to  the  difference  of 
potential  between  A  and  B  is  a  constant. 

It  is  important  to  realise  how  the  charge  on  B  is  distributed  in  this 
case.  Since  all  the  lines  of  force  from  A  end  on  B  and  since  the 
charges  at  the  two  ends  of  a  line  of  force  are  equal  and  opposite,  the 
charge  on  the  inner  surface  of  B  must  be  -  Ql ,  and  this  charge  resides 
exclusively  on  the  interfaces  between  the  conductor  and  the  dielectric, 
but  the  total  charge  on  B  is  Q2,  hence  the  charge  on  the  outer  surface 
of  B  must  be  Q\  +  Q2 .  We  may  also  consider  the  forces  due  to  these 
charges.  Within  A  the  potential  is  constant  and  equal  to  V1  and  the 
force  is  zero.  Between  A  and  B  the  potential  changes  from  Vl  to  F2, 
the  force  is  entirely  due  to  Q,.  At  the  inner  surface  of  B  and  at  all 
points  exterior  to  this,  the  force  and  potential  due  to  Ql  on  A  are  each 
equal  and  opposite  to  the  force  and  potential  due  to  -  Qj  on  B.  The 
effect  of  A  is  entirely  screened  by  the  effect  due  to  the  inner  surface 
of  B.  Exterior  to  this  inner  surface,  force  and  potential  are  both  due 
to  Qi  +  Qz  on  the  outer  surface  of  B.  At  points  within  this  outer 
surface,  points  that  is  within  the  substance  of  B,  the  potential  due  to 
the  charge  Q1  +  Qz  is  constant  and  the  force  is  therefore  zero.  At  all 
external  points  the  potential  and  the  force  are  those  which  arise  from 

+       on  the  outer  surface  of  B. 


62  ELECTRICITY  [CH.  IV 

Now  with  a  condenser  such  as  we  have  described  pre- 
viously —  two  flat  plates  near  together  —  the  above  conditions 
do  not  hold  ;  all  the  lines  of  force  from  A  do  not  pass  to  B. 
But  the  condition  is  very  nearly  realised  when  the  plates 
are  near  together,  the  number  of  lines  of  force  from  the  back 
of  the  plate  A  is  very  small,  almost  all  issue  from  its  front 
surface  and  fall  on  B. 

Hence  in  this  case  also  we  may  say  that  the  ratio  of  the 
charge  to  the  difference  of  potential  is  constant,  and  define 
this  ratio  as  the  capacity  of  the  condenser  thus  : 

DEFINITION.     The  Capacity  of  a  condenser  is  the  ratio  of( 
its  charge  to  the  difference  of  potential  between  its  plates. 

If  we  call  the  capacity  C  we  have 


where  Q  is  the  charge,  V  and  V  the  potentials. 

We  may  notice  that  if  V  be  zero  so  that  all  neighbouring 
conductors  are  at  zero  potential  we  have  Q  =  C  V,  the  same 
equation  as  previously,  the  two  definitions  are  consistent. 

41.  Energy  of  a  Charged  Condenser.  We  know 
that  the  energy  of  a  charged  conductor  is  \QV.  Hence  in 
the  case  of  the  condenser  in  which  we  have  one  plate  with 
charge  Q  and  potential  V  and  another  with  charge  —  Q  and 
potential  V  the  energy  is 


This  may  be  written  \Q  (  V-  V)  or  J  C  (  V  -  V'}\ 

l<f 
2C' 

42.  Inductive  Capacity.  EXPERIMENT  15.  To  shew 
that  the  capacity  of  a  condenser  depends  on  the  nature  of  the 
dielectric  between  its  plates. 

Arrange  the  condenser  AB  (Fig.  25)  so  that  the  plate  A  is 
connected  to  an  electroscope  while  B  is  earthed.  Take  a  plate 
of  glass  or  other  insulating  material,  making  sure  that  its 
surfaces  are  unelectrified  by  passing  it  over  a  flame,  and  place 
it  between  the  plates  of  the  condenser,  taking  care  not  to 


40-43]  CONDENSERS  63 

touch  the  insulated  plate1,  the  divergence  of  the  gold  leaves 
is  reduced ;  a  change  of  the  dielectric  has  increased  the 
capacity  of  the  condenser,  and  reduced  the  potential  of  the 
insulated  plate. 

DEFINITION.  The  ratio  of  the  capacity  of  a  condenser 
having  a  given  material  for  its  dielectric a  to  the  capacity  of  the 
same  condenser  with  air  for  the  dielectric  is  called  the  Specific 
Inductive  Capacity  of  the  Condenser. 

The  term  specific  inductive  capacity  has  been  already 
denned,  see  Section  27,  and  we  have  learnt  that  the  force 
between  two  charges  e,  e  at  a  distance  r  apart  in  a  medium  of 
inductive  capacity  K  is  ee '//ir2.  It  follows  from  theory  and 
can  be  shewn  directly  by  experiment  that  the  two  definitions 
are  consistent.  The  quantity  K  given  by  the  above  equation 
is  the  inductive  capacity  as  defined  by  the  definition  of  this 
Section. 

43.  Calculation  of  Capacity.  The  capacity  of  a 
condenser  does  not  depend  on  its  charge  or  its  potential,  but 
merely  on  its  size  and  shape  and  the  distance  apart  of  its 
surfaces.  We  can  calculate  it  for  the  plate  condenser  thus 
if  we  assume  a  relation  which  is  readily  proved  as  a  con- 
sequence of  the  inverse  square  law,  but  of  which  the  proof 
lies  rather  outside  our  limits. 

The  relation  is  that,  if  in  a  medium  of  inductive  capacity  K, 
R  be  the  resultant  force  just  outside  a  conducting  surface,  on 
which  the  surface  density  is  <r,  then  K.  R  =  ±TT(T,  or  in  words, 
the  product  of  the  resultant  force  and  the  inductive  capacity 
is  equal  to  4?r  times  the  surface  density.  This  is  known  as 
Coulomb's  law. 

Now  in  the  plate  condenser  if  the  distance  apart  of  the 
plates  is  small  compared  with  their  size  the  field  of  force  is 
uniform  and  the  force  is  constant.  Hence  if  c  be  the 


1  Theoretically  contact  with  the  insulated  plate  ought  not  to  matter, 
it  is  difficult  however  to  prevent  some  leakage  over  the  surface  of  the 
dielectric,  and  this  if  it  occurred  would  simulate  the  result  sought  for. 

2  In  this  definition  the   dielectric   is   supposed    to   replace   the   air 
completely. 


64  ELECTRICITY  [CH.  IV 

distance  apart  of  the  plates,  F,  V  their  potentials,  the  work 
done  in  carrying  a  unit  charge  across  is  Re  but  it  is  also 

V-  V. 
Hence  Rc  =  V-V, 

»     V~r 
or  R  =  . 

c 

But  if  Q  is  the  charge,  S  the  area  of  the  positive  plate  and 
C  the  capacity,  then 

G«=&r, 
also  by  Coulomb's  law       KR  =  47r<r. 

-47TO-          4:TrQ 

Hence  £=_==_. 


rru         t  V~V> 

Therefore  -=§  =  —      —  . 

AO  C 

„       Q        KS 
=  F-f'=-£S- 

Thus  if  we  know  the  area  and  distance  apart  of  the  plates 
and  the  inductive  capacity  of  the  dielectric  we  can  calculate 
the  capacity  of  the  condenser. 

If  C0  be  the  capacity  of  the  same  condenser  with  air 
as  dielectric,  since  for  air  K=l,  we  find 

s 


Hence  from  these  two  results  we  obtain 

£*7S-> 

and  this  agrees  with  the  definition  of  specific  inductive  capacity 
given  in  Section  27  above. 

44.  Leyden  Jar.  Condensers  take  various  forms. 
One  of  the  most  usual  is  that  of  the  Leyden  Jar.  This  as 
usually  constructed  is  shewn  in  Fig.  27.  A  wide-mouthed 
glass  bottle  is  coated  in  part  within  and  without  with  tinfoil. 
The  glass  above  the  tinfoil  is  coated  with  shellac  varnish  to 


43-44] 


CONDENSERS 


65 


Fig.  27. 


maintain  the  insulation ;  the  bottle  is  closed  by  a  wooden 
stopper  through  which  runs  a  piece  of 
brass  rod  ;  the  upper  end  of  this  carries 
a  knob  or  ball,  the  lower  end  is  in  con- 
nexion by  means  of  a  light  chain  or  a 
piece  of  wire  with  the  inner  coating  of 
tinfoil.  The  two  coatings  of  tinfoil  con- 
stitute the  plates  of  the  condenser ;  the 
glass  is  the  dielectric,  the  inner  coating 
is  insulated  by  means  of  the  glass,  the 
outer  coating  can  be  insulated  if  neces- 
sary by  placing  the  whole  instrument  on 
an  insulating  stand. 

To  charge  the  jar  the  knob  is  brought  near  to  an  electrical 
machine  or  other  source  of  electricity,  sparks  pass  across 
from  the  machine  to  the  jar  and  a  considerable  quantity  of 
electricity  can  be  communicated  to  the  inner  coating. 

If  a  connexion  be  made  between  the  two  coatings  by 
means  of  the  discharging  tongs  shewn  in 
Fig.  28,  one  ball  of  which  is  made  to  touch 
the  outer  coating  while  the  other  is  brought 
near  to  the  knob,  a  spark  passes.  To  secure 
the  observer  from  shock  the  handle  of  the 
tongs  is  of  glass  or  some  other  insulator. 

The  hygroscopic  qualities  of  various  glasses 
differ  greatly;  it  is  important  to  choose  for 
a  Leyden  Jar  a  glass  which  is  not  hygro- 
scopic. 

Another  form  of  condenser,  Fig.  29,  consists  of  a  number 
of  sheets  of  tinfoil  insulated  from  each  other  by  alternate 


Fig.  28. 


Fig.  29. 

sheets  of  mica  or  of  paraffin  paper.     The  odd  sheets  of  the 
tinfoil,  the  first,  third,  fifth,  etc.  are  connected  together  and 

5 


G.   E. 


66 


ELECTRICITY 


[CH.  IV 


form  one  plate  of  the  condenser,  the  even  sheets,  the  second, 
fourth,  sixth,  etc.  are  also  connected  and  form  the  other  ;  the 
whole  is  mounted  in  a  case,  the  two  plates  being  connected  to 
the  terminals  A  and  B. 

45.  Batteries .  of  Ley  den  Jars.  The  capacity  of 
a  condenser  is,  we  have  seen,  proportional  to  the  surface  of 
one  of  its  plates  and  inversely  proportional  to  the  distance 
between  them ;  it  can  be  increased  then  either  by  reducing 
this  distance  or  by  increasing  the  area  of  the  surface.  There 
are  limits  however  below  which  we  cannot  reduce  the  distance, 
for  if  the  dielectric  be  too  thin  it  is  pierced  by  an  electric 
spark  when  the  potential  difference  is  raised.  We  can  how- 
ever practically  increase  the  surface  by  connecting  up  a 
number  of  jars  as  shewn  in  Fig.  30.  The  outer  plates  are  all 


Fig.  30. 

connected  together  by  placing  the  jars  on  a  sheet  of  tinfoil  or 
metal ;  the  knobs  of  all  the  jars  are  in  electrical  communica- 
tion and  thus  the  inner  coatings  are  also  connected.  In  such 
an  arrangement  it  is  clear  that  the  capacity  of  the  whole  is 
the  sum  of  the  capacities  of  the  individual  jars. 

For  some  purposes  jars  are  connected  as  shewn  in  Fig.  31, 
they  are  then  said  to  be  "in  cascade."  The  outer  coating 
of  the  first  jar  is  connected  to  the  inner  coating  of  the  second 


44-45] 


CONDENSERS 


67 


and  so  on  in  succession.  All  the  jars  except  the  last  have 
their  outer  as  well  as  their  inner  coatings  insulated,  the  outer 
coating  of  the  last  jar  is  to  earth. 


Fig.  31. 

Suppose  a  charge  Q  given  to  the  inner  coating  of  the  first 
jar,  the  one  to  the  right  hand  in  the  figure.  Let  its  potential 
be  Fi  and  the  potentials  of  the  successive  inner  coatings  be 
F2,  F3  ...  Vn.  The  potential  of  the  outer  coating  of  the  nth 
or  last  jar  is  zero. 

The  charge  Q  induces  —  Q  on  the  outer  coating  of  the  first 
jar  and  repels  Q  to  the  inner  coating  of  the  second,  and  this 
is  continued  throughout  the  system.  Thus  the  charges  of  all 
the  jars  are  the  same.  Let  C^  C2  .. 


Cn  be  their  capacities. 


Then  we  have 


Q  = 


,-  V,\ 


Whence 


V      V  - 

*  1  ~    ¥  2  ~   /-»    J 

^1 
7      V  -Q- 

y  2       '3  —  n  ' 


5—2 


68 


ELECTRICITY 

Q 


[CH.  IV 


Fw-0  =  -^, 

Therefore  adding  these  all  together 

V          Oll     4-     1     4  H 

"1  =  dMTT  +  zr  +  —  +  7Tf  • 

\^l        v2  L>n) 

But  if  C  be  the  equivalent  capacity  of  the  system 

r,=l 

111  i 

c=c,  +  c2+-+c;- 

The  whole  system  is  equivalent  to  a  single  jar  having  a 

•    „  X«*«  1  .<••  ,• 

anim.r.iATi 


capacity  C  given  by  this  equation. 


46.  Experiments  with  Ley  den  Jars.     By  making 
the  coatings  of  a  Leyden  Jar  removable  it  oan  be  used  to  shew 
that  the  charge   of   a   conductor  resides 

on  the  surface  of  the  dielectric  which 
insulates  it.  This  was  first  done  by 
Benjamin  Franklin.  The  jar  as  shewn 
in  Fig.  32  is  in  the  form  of  a  tumbler; 
the  coatings  are  both  of  tin  or  brass  ; 
the  inner  coating  can  be  lifted  out  of 
the  glass  and  then  the  glass  can  be  re- 
moved from  the  outer  coating.  The  jar  is 
charged  in  the  usual  way,  on  removing 
the  coatings  and  examining  them  they 
are  found  to  be  uncharged;  the  glass 
when  examined  by  aid  of  an  electroscope 
is  found  to  be  strongly  charged. 

47.  The   Condensing   Electro- 
scope.    In  this  instrument,  Fig.  33,  the 
action  of  a  condenser  is  applied  to  render 
sensible  the  action  on  a  gold-leaf  or  other 
electroscope  of  a  source  of  electricity  at 
low  potential.   Such  a  source,  if  connected 
to  the  electroscope  directly,   is  not  suf- 

ficient to  produce  any  visible  effect  on  the  leaves. 


45-47] 


CONDENSERS 


69 


Instead  of  a  knob  the  electroscope  is  fitted  with  a  flat 
plate,  the  upper  side  of  which  is  varnished  so  as  to  make  it 
insulate.  An  insulating 
handle  is  attached  to  a 
second  similar  plate  which 
is  varnished  on  the  lower 
side.  On  placing  this  plate 
on  the  electroscope  the  two 
constitute  a  condenser  of 
large  capacity,  the  dielec- 
tric being  the  thin  layer  of 
varnish. 

The  upper  plate  is  now 
connected  to  the  electri- 
fied body  and  the  lower 
plate  is  earthed  for  a 
moment.  The  difference 
of  potential  between  the 
plates  is  not  large,  but  in 
consequence  of  their  large 
capacity  a  considerable 
charge,  of  the  opposite  sign  to  that  on  the  body,  is  communi- 
cated to  the  lower  plate  and  remains  there  when  the  earth 
contact  is  broken;  the  lower  plate  and  the  electroscope  are 
then  at  potential  zero.  Now  remove  the  upper  plate,  which 
for  the  moment  we  will  assume  to  be  positive,  in  consequence 
of  the  removal  of  the  positive  plate  the  potential  of  the  lower 
plate  with  its  negative  charge  falls  considerably  below  that 
of  the  cage  or  tinfoil  strips  on  the  glass  cover ;  the  leaves 
therefore  diverge  with  negative  electricity.  The  small  differ- 
ence of  potential  between  the  plates  which  exists  when  the 
condenser  has  a  large  capacity  is  increased  many  times  when 
the  upper  plate  is  removed  and  the  condenser  capacity  in 
consequence  reduced. 

By  means  of  this  condenser  action  the  gold-leaf  electro- 
scope may  be  used  to  indicate  very  small  differences  of 
potential. 


Fig.  33. 


CHAPTER  V. 

ELECTRICAL   MACHINES. 

48.  Frictional  .Machines.     The  only  means  of   ob- 
taining an  electric  charge  described  up   to  the  present  has 
been  by  the  friction   of    two   dielectric   materials.     Various 
forms  of  apparatus  have  been  devised  with  a  view  of  obtaining 
larger  charges  than  it  is  possible  to  do  by  the  friction  of  a  rod 
of  glass  or  of  ebonite. 

49.  The  Plate  Electrical  Machine.     This,  which 
is   shewn   in   Fig.    34,   consists   of  a    circular  plate  of   glass 
mounted  so  as  to  rotate  about  a  horizontal  axis.     Two  pairs 
of  cushions  rub  against  the  glass  at  opposite  extremities  of  a 
vertical   diameter,   and   two    U-shaped    insulated    conductors 
with  points  on  the  side  towards  the  glass  are  placed  to  collect 
the  electricity  produced  by  the  friction.    The  points  are  known 
as  the  combs. 

These  two  conductors  are  connected  together  and  also  to 
an  insulated  conductor  known  as  the  prime  conductor  of  the 
machine. 

Two  opposite  quadrants  of  the  plate  between  the  cushions 
and  the  combs  are  usually  covered  with  flaps  of  oiled  silk. 
The  cushions  are  of  wash-leather  stuffed  with  horsehair,  covered 
with  an  amalgam  of  tin  and  zinc  with  mercury  smeared  on  to 
them  with  a  little  lard  or  tallow.  To  secure  good  working  the 
cushions  should  be  earth- con  nee  ted.  The  connexion  afforded 
by  the  wood  framing  of  the  machine  is  usually  sufficient ;  in 
some  cases  however  contact  is  obtained  by  a  piece  of  chain  or 


48-49] 


ELECTRICAL   MACHINES 


by  strips  of  tinfoil.  The  plate  is  turned  by  a  handle,  the 
direction  of  motion  past  the  cushions  being  from  the  uncovered 
towards  the  covered  quadrants. 


Fig.  34. 

When  the  machine  is  in  action  the  friction  of  the  cushions 
produces  positive  electrification  on  the  glass,  negative  on  the 
amalgam  ;  the  negative  escapes  to  the  earth,  the  positive  is 
carried  forward  on  the  glass  towards  the  combs.  As  this 
positive  electricity  approaches  the  combs  the  prime  conductor 
becomes  electrified  by  induction  ;  negative  electrification  is 
attracted  to  the  combs  and  positive  repelled  to  the  other  end 
of  the  conductor.  The  electrical  force  at  the  points  becomes 
very  great  and  a  wind  of  negatively  electrified  particles  of  air 
blows  from  them  on  to  the  glass ;  the  glass  plate  is  thus 
discharged  ;  its  electrification  has  passed  to  the  prime  con- 
ductor ;  the  unelectrified  glass  becomes  charged  again  as  it 
passes  through  the  cushions  and  the  process  is  repeated. 


72  ELECTRICITY  [CH.  V 

The  silk  flaps  attached  to  the  cushions  become  negatively 
electrified,  and  by  their  action  tend  to  produce  a  more  uni- 
form slope  of  the  potential  along  the  surface  of  the  glass  than 
would  otherwise  be  possible ;  the  charge  on  the  glass  is  thus 
prevented  from  passing  back  to  the  rubber  and  the  difference 
of  potential  between  the  rubber  and  the  combs  is  increased. 
Sparks  can  be  collected  from  the  prime  conductor  by  bringing 
another  conductor  up  to  it. 

There  are  various  forms  of  electrical  machines.  The 
principle  however  is  much  the  same  for  all.  Various  pre- 
cautions have  to  be  taken  in  working  them.  Glass  is 
hygroscopic ;  it  is  necessary  therefore  that  all  glass  surfaces 
should  be  dry  and  clean ;  it  is  a  good  plan  to  coat  the  glass 
insulating  stems  with  a  thin  layer  of  varnish  made  by  dis- 
solving shellac  in  pure  alcohol ;  before  use  it  is  desirable  to 
warm  the  machine  slightly.  In  consequence  chiefly  of  these 
various  defects  frictional  machines  are  now  practically  ob- 
solete ;  they  have  been  superseded  by  influence  machines. 

5O.  The  Electric  Spark.  Various  experiments 
already  described  can  be  performed  on  a  larger  scale  with 
an  electrical  machine.  If  a  conductor  be  brought  near  to  a 
machine  which  is  being  worked,  the  air  between  the  machine 
and  the  conductor  is  subjected  to  a  gradually  increasing 
electrical  stress,  and  after  a  time  its  insulating  power  is  over- 
come and  it  gives  way,  a  spark  passes  from  the  machine  to  the 
conductor.  The  light  from  the  spark  is  due  to  the  very 
great  rise  in  temperature  produced  by  its  passage,  the  air  is 
hereby  rendered  incandescent.  As  the  machine  is  worked 
sparks  continue  to  pass  until  the  potential  of  the  conductor 
has  risen  greatly  and  has  approached  so  nearly  to  that  of  the 
machine  that  the  difference  between  the  two  is  insufficient  to 
rupture  the  air. 

When  the  conductor  of  the  machine  has  been  raised  to  a 
high  potential  and  the  machine  continues  to  be  worked,  the 
electricity  is  discharged  into  the  air;  the  air  particles  near 
the  machine  are  charged  and  the  forces  on  them  become  very 
great;  the  particles  are  repelled  from  the  machine  to  have 
their  places  taken  by  others  and  an  electric  wind  is  set  up. 


49-50] 


ELECTRICAL   MACHINES 


In  the  dark  the  positive  conductor  of  the  machine  is  sur- 
rounded with  a  violet  glow  which  is  specially  marked  near 
any  angle  or  sharp  point.  To  prevent  this  loss  all  points  or 
angles  should  be  avoided  as  far  as  possible.  Any  points  on 
the  negative  conductor  shew  stars  of  white  light. 

The  effect  of  points  in  discharging  electricity  may  be 
illustrated  in  various  ways.  Thus  with  a  given  machine  in 
its  normal  state  it  may  be  possible  to  draw  sparks  an  inch  or 
more  in  length  ;  if  a  fine  point  be  affixed  to  the  conductor  the 
length  of  spark  is  reduced  to  a  small  fraction  of  what  it  was 
previously  and  the  violet  glow  of  the  positive  discharge,  is 
seen  from  the  point.  If  a  lighted  candle  be  held  as  in 
Fig.  35  near  the  point,  the  flame  is  violently  blown  away. 


Fig.  35. 

The  electric  windmill  is  another  example  of  the  action  of 
points.  A  number  of  pins  with  their  points  turned  in  the 
same  direction  are  attached  like  the  spokes  of  a  wheel  to  a 
small  central  cup  and  balanced  on  a  pivot  connected  with  the 
conductor  of  an  electrical  machine ;  on  working  the  machine 
the  pins  revolve,  moving  round  in  the  opposite  direction  to 
that  in  which  the  points  are  bent.  The  electric  wind  which 
blows  from  each  point  reacts  on  the  mill  and  drives  it  round. 
If  the  points  be  blunted  by  sticking  on  to  each  a  bit  of  sealing- 
wax  the  mill  will  no  longer  turn. 

More  or  less  successful  attempts  have  been  made  to  utilize 
this  electric  wind  on  a  large  scale  to  remove  metallic  fumes  or 


ELECTRICITY 


[CH.  V 


smoke  from  the  air.  If  the  air  near  a  discharging  point  is 
loaded  with  small  particles,  these  are  driven  away  and  deposited 
on  opposing  surfaces. 

51.     The  Electrophorus.     The  instrument,  Fig.  36, 
consists  of  a  flat  smooth  plate  of  resin,  ebonite  or  some  other 
insulating  substance — the  cake 
— which  rests  on  a  metal  plate 
— the  sole — and   of   a   second 
metal  plate — the  cover — which 
is  supported  by  an  insulating 
handle.     To  use  the  instrument 
the  cake  is  electrified  by  friction 
with  catskin,  receiving  hereby 
a  negative  charge ;  the  cover  is 
laid  on  it  and  touched  by  the 
experimenter  for  a  moment ;  it 
is  then  removed  by  the  insu- 
lating handle  and  will  be  found 
to  be  positively  electrified;  if 
it  be  discharged  and  again  laid 
on  the  cake  a  second  positive 
charge  can  be  obtained  with- 
out  renewed    use   of    the    catskin,    and    this    can   in   a   dry 
ftmosphere  be  repeated  many  times. 
We  may  explain  the  action   thus.      When   the  cover  is 
placed  on  the  electrified  cake  it  rests  on  a  few  points  only, 
elsewhere  the  two  are  separated  by  a  thin  layer  of  air.     The 
electrification  of  the  cake  acts  inductively  on  the  cover.    When 
the  cover  is  touched  positive 
electricity  passes  on  to  it  from 
the  finger,  the  outer  field  is 
destroyed   and    the   cover    is 
left   with   a   positive    charge. 
The  distribution  is  as  shewn 
in  Fig.  37.     On  removing  the 
cover  this  positive  charge  is    _ 
carried     away    with    it,    the 
cake     is     left,      except     for 
leakage,  with  practically  the 
same    charge    as    it    possessed    previously,     and    the    cover 


Fig.  36. 


(4- 


-h   H-   +  -K) 


Fig.  37. 


50-51]  ELECTRICAL   MACHINES  75 

after    discharge    can    be    replaced    and    the    operations    re- 
peated. 

The  action  of  the  sole  tends  to  reduce  the  leakage  into 
the  air  and  thus  to  prolong  the  period  during  which  the 
effect  of  a  single  application  of  the  catskin  is  effective.  For 
the  sole  being  in  contact  with  the  earth  is  necessarily  at  zero 
potential;  the  strongly  electrified  ebonite  would  if  the  sole 
were  not  present  be  at  a  considerable  negative  potential,  and 
the  force  tending  to  discharge  it  would  be  large ;  the  presence 
of  the  •  sole  reduces  the  difference  of  potential  between  the 
cake  and  the  air  near  it  and  thus  reduces  the  tendency 
to  leak. 

It  is  interesting  to  trace  the  distribution,  of  the  lines 
of  force  during  these  various  processes.  When  the  cake 
receives  its  negative  charge  the  sole  becomes  positive.  Lines 
of  force  pass  from  it  through  the  ebonite  to  the  upper  surface 
of  the  cake  ;  the  field  is  almost  entirely  confined  to  the  ebonite, 
though  a  few  lines  may  pass  from  the  table  and  walls  of  the 


/^~\ •^TN  ^-/-rr-rY-fs. 

f  r-rrTTTTrrfA          r   r  r  r  i 

Pig.  38.  Fig- 


r  T  T  T  T  n 


Fig.  40.  Fig.  41. 

Figs.   38,  39,  40,  41. 

room  directly  into  the  cake.  This  is  shewn  in  Fig.  38.  The 
effect  of  bringing  the  cover  near  is  illustrated  in  Fig.  39. 
Lines  of  force  pass  from  the  cover  to  the  cake;  the  field 


76 


ELECTRICITY 


[CH.  V 


is  transferred  in  great  measure  from  the  ebonite  to  the  air- 
space between  the  cake  and  the  cover,  but  the  upper  surface 
of  the  cover  being  negatively  charged  receives  from  the  walls 
and  surrounding  objects  lines  of  force  equal  in  number  to 
those  which  pass  from  it  to  the  cake;  the  number  of  lines 
traversing  the  ebonite  is  small  and  the  positive  charge  of 
the  sole  is  now  distributed  over  the  walls,  there  is  an  external 
field  due  to  this  charge  and  the  negative  charge  on  the  upper 
side  of  the  cover.  When  the  cover  is  touched  this  external 
field  is  destroyed,  the  field  in  the  air-gap  between  the  cake 
and  the  cover  remaining  much  as  before,  and  we  have  the 
condition  shewn  in  Fig.  40. 

As  the  cover  is  removed  the  lines  between  it  and  the  cake 
lengthen,  some  of  them  bulge  outwards  as  shewn  in  Fig.  41, 
and  finally  come  into  contact  with  the  walls  ;  here  they  break 
into  two  parts,  one  of  which  shortens  again  until  it  passes 
directly  through  the  ebonite  between  the  sole  and  the  cake, 
while  the  other  ends  in  a  negative 
charge  on  the  walls.  This  is  shewn  in 
Fig.  42.  As  the  cover  is  moved  further 
away  this  happens  to  an  increasing 
number  of  the  lines  between  it  and 
the  cake,  until  finally  they  are  practi- 
cally all  broken  and  we  are  left, 
so  far  as  the  cake  is  concerned,  with 
the  same  state  of  affairs  as  after  the 
application  of  the  catskin,  but  in  addi- 
tion there  is  now  another  field  due  to 


Fig.  42. 


the  charge  on  the  cover.  This  is  shewn  in  Fig.  43.  Electric 
energy  is  stored  in  this  field,  and  this  energy  is  derived  from 
the  additional  work  which  has 
been  required  to  lift  the  cover 
from  the  cake  in  consequence 
of  its  positive  charge. 

Various  mechanical  ar-  r_^_^_____ 
rangements  might  be  devised  I  T  T  T  T  I 
to  carry  out  the  operations  -pig  43 

required  to  charge  a  body  by 

the  electrophorus,  and  thus  to  give  us  a  machine  producing  by 
induction  a  continuous  supply  of  electricity.     This  end  how- 


51-53]  ELECTRICAL   MACHINES  77 

ever  is  attained  more  easily  by  means  of  some  of  the  induction 
machines  we  are  about  to  describe. 

52.  Influence  Machines.  The  principle  of  all  these 
is  much  the  same.  In  their  simplest  form  there  are  two 
fixed  insulated  conductors  A  and  B  which  we  may  call 
the  collectors,  and  a  moving  conductor  (7,  which  can,  after 
being  electrified,  be  brought  near  to  A  and  B  in  turn  in  such 
a  position  that  if  contact  be  established  or  a  spark  allowed 
to  pass,  electricity  will  be  transferred  from  C  to  A  or  B  as  the 
case  may  be,  and  G  will  move  on  unelectrified. 

Suppose  now  that  we  start  with  a  slight  difference  of 
potential  between  A  and  B,  A  being  say  slightly  positive  and 
B  negative.  Bring  C  near  to  A  and  connect  it  to  earth ; 
it  is  electrified  negatively  by  induction.  Break  the  earth 
connexion  and  move  C  on  until  it  is  near  to  B.  Connect 
B  and  (7,  the  negative  charge  on  C  passes  to  B,  which  becomes 
more  negative.  Break  this  connexion,  and  while  C  is  still 
near  to  B  connect  it  to  earth ;  it  receives  a  positive  charge  by 
induction.  Break  this  earth  connexion  and  bring  C  near  to  A, 
putting  the  two  into  connexion  •  the  positive  charge  passes 
to  A  which  becomes  more  positive.  We  have  thus  come  to  the 
end  of  a  cycle  ;  on  breaking  the  connexion  between  A  and  C 
and  again  putting  C  to  earth,  it  receives  for  a  second  time 
a  negative  charge  and  the  whole  series  can  be  repeated.  But 
we  must  notice  that  this  second  negative  charge  is  greater 
than  the  first,  for  by  the  action  of  the  machine  A  has  received 
an  increase  in  its  positive  charge  and  therefore  induces  a 
larger  charge  on  C  when  near  it  than  it  did  at  first.  Thus 
not  only  is  the  potential  difference  between  A  and  B  con- 
tinually increased  by  the  action  of  the  machine,  but  also 
the  rate  at  which  it  is  increased  grows  rapidly. 

Nicholson's  revolving  doubler,  invented  in  1788,  was  pro- 
bably the  first  machine  of  this  kind;  since  that  time  it  has 
taken  various  forms,  and  of  these  we  will  describe  one  or  two. 

53.  The  Replenisher.  In  this  instrument  (Figs.  44 
and  45),  designed  by  Sir  William  Thomson  (Lord  Kelvin)  in 
1867,  the  two  collectors  A  and  B  are  two  portions  of  a  cylinder, 
the  axis  of  which  cuts  the  paper  at  right  angles  at  0.  Each  of 


78 


ELECTKICITY 


[CH.  V 


these  is  insulated,  and  from  the  inside  of  each  there  projects 
a  small  spring,  a,  b.  There  are  two  carriers  C19  C2,  which  are 
portions  of  a  second  smaller  cylinder.  These  are  connected 


Fig.  44. 


Fig.  45. 


together  by  an  insulating  bar  carried  by  an  ebonite  rod  which 
can  be  made  to  turn  about  the  axis  of  the  outer  cylinder. 
Two  springs  cly  d^  connected  together  by  a  wire  which  need 
not  be  insulated,  touch  the  carriers  when  these  are  in  a 
position  to  be  affected  by  the  inductive  action  of  charges  on 
A  and  B.  As  the  carriers  turn  still  further  in  the  direction 
of  the  arrow,  contact  with  clt  dl  is  broken,  and  contact  with  6, 
a  is  made. 

Suppose  now  that  there  is  a  small  initial  difference  of 
potential  between  A  and  £,  produced  if  need  be  by  electri- 
fying A  by  induction  from  an  ebonite  rod  so  that  A  is  positive 
with  respect  to  B. 

Then  negative  electricity  is  induced  on  the  carrier  C1,  and 
positive  is  repelled  to  Cz.  The  carriers  are  so  placed  that 
a  motion  in  the  direction  of  the  arrow  from  the  position 
shewn  breaks  the  contacts  with  Cj  and  dlt  the  positive 
and  negative  charges  then  are  insulated  from  each  other,  the 
negative  charge  on  Cl  is  carried  round  towards  B,  the  positive 
on  (72  towards  A.  When  Cl  is  in  contact  with  b  it  is  approxi- 
mately surrounded  by  the  conductor  B  and,  a  large  fraction 
of  its  negative  charge  passes  to  the  outside  of  that  conductor ; 


53-54] 


ELECTRICAL    MACHINES 


79 


the  converse  happens  to  C2.  As  the  motion  continues  Cl 
comes  into  contact  with  d-^  and  (72  with  ca ;  in  consequence  of 
the  inductive  action  of  A  and  B,  the  carrier  Cl  becomes  positive 
and  (?2  negative,  and  the  process  is  repeated.  The  difference  of 
potential  between  A  and  B  is  thus  continually  increased  and 
that  at  an  increasing  rate.  A  conductor  connected  to  A 
becomes  increasingly  positive,  one  connected  to  B  increasingly 
negative,  and  this  continues  until  the  insulation  gives  way 
somewhere  and  a  spark  passes.  The  instrument  in  this  form, 
however,  is  not  used  to  produce  large  charges  of  electricity, 
but  rather  to  maintain  some  portion  of  another  instrument 
at  a  given  potential.  As  the  potential  falls  by  leakage  it  is 
raised  by  giving  the  replenisher  a  few  turns. 


Fig.  46. 

54.  Wimshurst's  Machine.  As  an  example  of  a 
powerful  influence  machine  we  will  describe  Wimshurst's. 
This  is  shewn  in  Fig.  46. 

It  consists  of  two  plates  of  glass  which  carry  narrow  strips 
of  tin-foil  arranged  radially  at  equal  distances  apart.  The 


80 


ELECTRICITY 


[CH.  V 


plates,  which  are  coated  with  shellac  to  preserve  their  insulating 
power,  rotate  in  opposite  directions  about  a  horizontal  axis. 
At  the  opposite  ends  of  the  horizontal  diameter  are  two 
U-shaped  conductors  furnished  with  points  on  the  sides 
towards  the  plates ;  each  of  these  is  insulated  and  connected 
with  a  Leyden  jar  or  other  condenser.  (See  Section  44.) 

Discharging  rods  fitted  with  insulated  handles  are  attached 
to  the  conductors.  Two  diametral  conductors  which  need  not 
be  insulated  are  placed  as  shewn  in  the  figure,  being  approxi- 
mately at  right  angles  to  each  other.  These  carry  brushes, 
which  just  touch  the  tin-foil  strips  as  they  pass  under  them. 


Fig.  47. 

The  action  of  the  machine  is  best  explained  by  the  diagram 
Fig.  47,  in  which,  following  a  suggestion  due  to  Prof.  S.  P. 
Thompson,  the  rotating  plates  are  represented  as  though  they 
were  two  cylinders  of  glass  rotating  in  opposite  directions. 
The  tin-foil  sectors  are  shewn  as  dark  lines  on  the  surface  of 


54]  ELECTRICAL   MACHINES  81 

the  cylinders,  the  diametral  conductors  pq,  rs  occupy  the 
position  indicated,  and  the  cylinders  rotate  in  the  directions 
of  the  arrows  ;  the  inner  cylinder,  which  corresponds  to  the 
front  plate  of  Figure  46,  having  a  right-handed,  and  the  outer 
cylinder  a  left-handed  motion.  Now  suppose  that  one  of  the 
strips,  P,  on  the  upper  side  of  the  outer  cylinder  becomes 
positively  electrified ;  as  it  rotates  it  will  be  brought  opposite 
to  a  strip  P'  on  the  inner  cylinder  at  the  moment  when  this 
is  in  contact  with  the  brush  p.  This  strip  then  acquires  a 
negative  charge,  the  strip  Q'  in  contact  with  q  being  simul- 
taneously charged  positively ;  as  the  rotation  continues  the 
induced  negative  charge  on  P'  is  carried  towards  the  right-hand 
comb  #,  the  positive  charge  on  Q'  towards  the  left-hand  comb 
A.  Consider,  however,  the  original  strip  P,  in  its  motion  it  is 
brought  under  the  left-hand  comb  A.  A  negative  wind  blows 
from  the  comb  discharging  P,  and  A  acquires  a  positive  charge 
while  P  moves  on  uncharged.  Meantime  P'  with  its  negative 
charge  and  Q'  with  its  positive  have  been  brought  opposite 
to  strips  at  R',  S',  which  are  connected  together  by  the  second 
diametral  conductor  rs.  R'  hereby  acquires  a  negative  charge 
and  S'  a  positive  one  which  are  carried  on  towards  B  and  A 
respectively,  while  P'  and  Q'  continue  their  motions.  As  P' 
comes  under  the  comb  B  a  positive  wind  is  produced  from  the 
points  and  B  receives  a  negative  charge,  A  at  the  same  time 
receiving  a  positive  charge  from  Q'.  R',  S'  again  in  their 
motion  have  been  brought  opposite  to  the  diametral  conductor 
pq,  and  after  inducing  charges  on  the  strips  then  in  contact 
with  this  conductor  have  passed  on  to  deliver  their  charges 
to  B  and  A.  In  this  way  the  difference  of  potential  between 
B  and  A  is  continually  increased. 

It  will  be  noticed  that  the  upper  part  of  the  outer  cylinder 
is  always  carrying  positive  electricity  from  r  to  A,  while  its 
lower  part  carries  negative  from  s  to  B.  Similarly  the  upper 
part  of  the  inner  cylinder  carries  negative  from  p  to  B, 
the  lower  part  carries  positive  from  q  to  A. 

In  working  with  a  Wimshurst  machine  it  is  not  usually 
necessary  to  charge  one  of  the  strips  to  start  with,  the  slight 
friction  between  the  brushes  and  the  strips,  or  the  small 
residual  charge  retained  from  previous  use,  is  generally 
sufficient  to  start  the  machine. 


G.  E. 


6 


82 


ELECTRICITY 


[CH.  V 


The  object  of  the  Ley  den  jars  is  to  increase  the  quantity 
of  electricity  given  by  each  spark  ;  the  sparks  pass,  for  a  given 
position  of  the  knobs,  when  the  potential  difference  between 
them  exceeds  a  certain  limiting  value.  Now  the  Leyden  jars 
increase  the  capacity  of  the  conductors,  they  increase  therefore 
the  quantity  which  is  required  to  charge  them  to  a  given 
difference  of  potential.  Each  strip  as  it  passes  the  combs 
conveys  a  definite  charge,  and  it  will  need  the  passage  of  a 
larger  number  of  strips  to  produce  a  spark  if  the  jars  are  on 
than  is  required  when  they  are  off.  Thus9  the  sparks  occur 
less  frequently  with  the  jars  on  than  without  them.  But  at 
each  spark  the  conductors  are  completely  discharged  and  the 
charges  are  increased  by  connecting  up  the  jars ;  each  spark 
contains  a  larger  quantity  of  electricity  than  would  pass  if  the 
jars  were  removed. 

In  large  machines  several  pairs  of  plates  are  mounted 
together  on  the  same  axis. 


Fig.  48. 

55.     Holtz    Machine.     The    Holtz   Machine   is  older 
than  the  Wimshurst,  and  in  consequence  of  its  liability  to 


54-55]  ELECTRICAL   MACHINES  83 

fail  in  action  in  damp  weather  is  less  used  at  present ;  a  brief 
description  therefore  will  suffice. 

The  machine  is  shewn  in  Fig.  48.  It  consists  of  two 
glass  plates,  one  fixed,  the  other  capable  of  rotation  about  a 
horizontal  axis.  At  opposite  extremities  of  a  diameter  of  the 
fixed  plate  two  holes  or  "windows"  are  cut  in  the  glass,  and 
two  pieces  of  varnished  cardboard — a  bad  conductor — are 
fastened  to  the  glass  on  the  side  remote  from  the  moving  plate. 
Each  of  these  cardboard  armatures  is  provided  with  a  point  or 
tongue,  which  projects  through  the  window,  so  as  almost  to 
graze  the  moving  plate ;  the  one  tongue  points  upwards,  the 
other  downwards.  Opposite  to  these  points,  but  separated  from 
them  by  the  moving  plate,  are  two  combs  connected  to  the 
conductors  of  the  machine.  These  can  be  placed  in  contact  by 
means  of  the  brass  rod  fitted  with  an  insulating  handle  shewn 
in  the  figure.  The  machine  is  not  self  starting ;  to  excite  it  a 
charged  piece  of  ebonite  is  held  near  one  of  the  armatures, 
and  the  conductors  brought  into  contact.  The  moving  plate 
is  then  set  into  rotation,  the  direction  of  motion  being  towards 
the  points,  and  after  a  little  time  a  peculiar  hissing  sound  is 
heard ;  while  the  combs  become  slightly  luminous.  The 
ebonite  may  now  be  withdrawn,  and  on  separating  the  two 
conductors  sparks  will  pass  between  them. 

The  action  of  the  machine  can  best  be  explained  by  the 
diagram  (Fig.  49)  in  which  the  moving  plate  only  is  repre- 
sented. P  and  Q  are  the  armatures,  A  and  B  the  conductors. 
When  the  negatively  charged  ebonite  E  is  brought  near,  the 
conductors  being  connected  as  at  a,  b,  positive  electricity 
is  induced  on  the  positive  comb  A,  and  negative  is  repelled 
to  B.  In  consequence  of  the  action  of  the  points  these 
two  electrifications  are  discharged  on  to  the  glass,  and 
the  moving  plate  becomes  positively  electrified  near  A, 
negatively  electrified  near  B,  and  this  continues  as  the 
plate  is  turned  until  the  whole  of  its  upper  half  is  posi- 
tively and  the  whole  of  its  lower  half  negatively  electrified. 
As  the  motion  continues  the  positively  charged  portion  of 
the  plate  comes  close  to  the  tongue  Q,  while  the  negatively 
charged  part  approaches  P.  The  armature  Q  therefore  dis- 
charges negative  electricity  on  to  the  back  of  the  plate, 

6—2 


84  ELECTEICITY  [CH.  V 

receiving  itself  a  positive  charge,  while  P  becomes  negative. 
These  charges  on  the  armatures  now  act  on  the  combs  in 
the  same  manner  as  the  original  charge  on  the  ebonite  did, 


Fig.  49. 

and  the  action  continues ;  moreover  since  the  charges  on  P 
and  Q  are  continually  increasing  the  potential  difference 
between  the  combs  rises  rapidly,  and  if  the  knobs  a,  b  are 
separated  it  may  be  sufficient  to  produce  a  spark  between 
them,  and  so  permit  of  the  transference  of  positive  electri- 
fication from  A  to  B  which  is  necessary  for  the  action  of  the 
machine.  Leyden  jars  are  sometimes  attached  to  the  machine, 
and  they  act  as  in  the  Wimshurst.  Sparks  pass  less  fre- 
quently, but  the  quantity  of  electricity  conveyed  in  each 
spark  is  increased.  If  the  knobs  are  separated  so  far  that 
the  sparks  cease  to  pass,  the  necessary  transfer  from  A  to  B 
cannot  take  place,  negative  electricity  is  carried  round 
on  the  glass  to  the  positive  armature,  while  positive  reaches 
the  negative  armature,  the  charges  on  the  armatures  are 
reduced,  they  may  even  change  sign,  and  the  machine  may 
cease  to  work ;  if  however  the  knobs  are  put  together  again, 
so  that  the  transfer  can  go  on,  the  usual  action  is  resumed.  It 
may  however  happen  that  by  this  the  poles  are  reversed,  for  if 
the  knobs  have  been  separated  too  long  the  negative  armature 
may  have  become  positive  and  the  positive  negative. 

Various  modifications  of  the  machine  have  been  introduced 
with  a  view  to  prevent  this  action,  but  into  these  it  is  not 
necessary  to  enter. 


55-57]  ELECTRICAL   MACHINES  85 

56.  The    Voss    Machine.      This    machine    acts    in 
the    same   manner  as  the   Replenisher  shewn  in  Figure  44. 
The  collectors  A,    B  are  replaced   by  two  pieces  of  tin-foil 
cemented  to  the  back   of   a  fixed  disc,  the  carrier  is  repre- 
sented by  a  number  of   studs    on    the    opposite    sides    of   a 
movable  disc.     To  the  springs   a,  b  correspond  two  brushes 
secured  to  the  collectors,  and  arranged  to  make  contact  with 
the  carriers  as  they  pass  round.     The   springs  c^  are  re- 
presented by   a   diametral  conductor,   with  brushes    of   thin 
brass  wire  which  connect  together  two  opposite  studs   when 
they  are  under  the  influence  of  the  collectors.     The  electricity 
produced  passes  from  the  collectors  through  the  points  on  the 
combs  to  the  conductors  of  the  machine.     The  action  is  the 
same  as  in  the  case  of  the  Replenisher,  the  friction  between 
the  studs  and  the  brushes  is  usually  sufficient  to  produce  some 
slight  potential  difference  between  the  collectors,  suppose  A  is 
positive.      A    stud    Cl    when    influenced    by    A    is   connected 
through   the  diametral  conductor  with  an  opposite  stud  (72. 
Thus  Cl  becomes  negative,  (72  positive ;  as  the  disc  rotates  Cl 
comes  in  contact  with  b,  and  part  of  its  negative  charge  passes 
to  B ;  when  it  again  reaches  the  diametral  conductor  (72,  it  is 
under  the  inductive  influence  of  B  and  becomes  positive,  and 
this  positive  charge  is  communicated  in  part  to  A  through  the 
brush  a.     Thus  the  difference  between  A  and  B  is  continually 
increased. 

57.  The    Water-dropping     Accumulator.      As 

another  example  of  an  influence  machine  Lord  Kelvin's  water- 
dropping  accumulator  (Fig.  50)  may  be  mentioned. 

It  consists  of  an  insulated  hollow  cylinder  A,  which  we 
will  suppose  to  be  charged  positively.  A  metal  pipe  B  con- 
nected to  earth,  having  a  fine  nozzle,  projects  down  the  axis  of 
this  cylinder  to  about  its  centre ;  below  is  a  second  insulated 
hollow  cylinder  C,  also  of  metal,  containing  a  funnel.  Water 
is  allowed  to  drop  slowly  from  the  nozzle  into  the  funnel,  from 
which  it  escapes.  Each  drop  on  leaving  the  nozzle  is  negatively 
electrified  owing  to  the  inductive  influence  of  A.  The  drop 
carries  with  it  to  the  funnel  its  negative  electrification  :  when 
in  contact  with  the  funnel  it  is  practically  in  the  interior  of  a 
hollow  closed  conductor,  it  gives  up  its  charge  to  the  cylinder  G 


86  ELECTRICITY  [CH.   V 

and  escapes  unelectrified.  This  process  could  go  on  continuously, 
except  for  the  fact  that  the  moisture  of  the  air  would  soon 
destroy  the  electrification  of  A,  but  by  putting  up  the  apparatus 


Fig.  50. 

in  duplicate,  and  connecting  A  to  C",  A'  to  C,  if  we  start  as 
before  with  A  positive,  A'  will  receive  a  negative  charge 
from  C.  This  will  act  inductively  on  the  water  dropping 
through  it,  thus  C'  will  become  positive,  some  of  the  positive 
charge  received  by  C'  will  pass  to  A,  and  as  a  result  the 
difference  between  A  and  A'  will  go  on  increasing. 


CHAPTER   VI. 


MEASUREMENT   OF   POTENTIAL   AND 
ELECTRIC   FORCE. 

58.  Electroscopes  and  Electrometers.     We  have 
described  various  simple  forms  of  apparatus  for  indicating  the 
presence  of  electricity,  such  as  the  pith  ball,  or  the  gold-leaf 
electroscope ;    we    must    now    consider    some    more    delicate 
appliances    which    can   be   used    for    measurement,    and    not 
merely  as  indicators  of  potential  difference. 

59.  Coulomb's  Torsion  Balance.     The  earliest  such 
instrument  was   Coulomb's  torsion  balance.     This,  which  is 
shewn  in  Fig.  51,  consists  of  a  small  gilded  pith  ball  A,  at  the 
end   of  a  long  light  rod   or  finger  ABC  of  some  insulating 
material ;  Coulomb  used  shellac.    This  rod  is  suspended  within 
a  glass  case  in  a  horizontal  position  by  means  of  a  very  tine 
vertical  wire  BD.     The  wire  passes  up  a  vertical  glass  tube 
attached  to  the  case  of  the  instrument,  and  is  fixed  at  D  to 
a  torsion  head.     The  tube  carries  a  horizontal  circular  plate 
at  its  upper  end,  the  edge  of  this  plate  is  graduated,  and  the 
torsion    head,    which    can    be    turned    about   a    vertical    axis 
through  the  centre  of  the  plate,  has   a  pointer  attached   so 
that  its  position  can  be  read  off  on  the  circle.    The  case  of  the 
instrument  is  usually  circular,  and  the  position  of  the  balance 
ball  can  be  observed  by  the  aid  of  a  graduated  circle  engraved 
on  the  case.     A  second  small  ball  E  carried  on  an  insulated 
stem  can  be  introduced  through  a  hole  in  the  case  and  brought 
into  any  desired  position  near  A. 


88 


ELECTRICITY 


[CH.  VI 


If  this  ball  E  be  charged  and  introduced  near  A  attraction 
at  first  takes  place ;  A  then  receives  a  portion  of  the  charge 
on  E  and  is  repelled ;  this  produces  a  twist  in  the  wire,  which 


Fig.  51. 

is  resisted  by  its  elasticity,  and  A  comes  to  rest  in  a  position 
in  which  the  electrical  force  of  repulsion  is  equal  to  the  force 
arising  from  the  twist  of  the  wire.  But  this  latter  force  is 
known  to  be  proportional  to  the  angle  through  which  the  wire 
is  twisted,  and  this  latter  angle  is  given  in  terms  of  the 
graduations  on  the  case  by  observing  the  original  and  final 
positions  of  A ;  if  then  we  find  that  in  one  experiment  the 
angle  of  twist  is  a°  and  in  another  /3°,  we  know  that  the  forces 
are  in  the  ratio  of  a  to  /3.  It  was  with  an  instrument  of  this 
kind  that  Coulomb  proved  the  law  of  the  inverse  squares, 
and  made  very  many  other  important  measurements.  Not 
the  least  interesting  among  the  historical  exhibits  in  the 
Paris  Exhibition  of  1900  was  Coulomb's  original  torsion 
balance. 

The  torsion   balance  has  now  been   superseded   by  more 


59]  MEASUREMENT   OF   POTENTIAL  89 

delicate  instruments,  but  the  general  idea  of  its  use  to  verify 
the  law  of  inverse  squares  may  be  given  thus.  Suppose  that 
in  the  above  experiment  the  shellac  carrier  supporting  the 
ball  A  is  twisted  through  an  angle  a° ;  the  distance  between 
the  balls  A  and  E,  if  E  be  placed  in  the  position  originally 
occupied  by  A,  will  be  approximately1  proportional  to  a. 

Now  by  twisting  the  torsion  head  in  the  opposite  direction 
to  that  in  which  A  has  moved,  the  distance  between  the  balls 
can  be  reduced.  Let  us  reduce  the  angular  distance  to  Ja, 
half  its  previous  value,  and  suppose  that  in  order  to  do  this 
we  have  to  turn  the  torsion  head  through  an  angle  J3.  The 
total  twist  on  the  wire  is  then  ft  +  J  a,  for  the  upper  end  has 
been  twisted  through  an  angle  /?,  while  the  lower  end  was 
turned  in  the  opposite  direction  through  an  angle  a  and  then 
brought  back  through  Ja. 

Thus  the  twist  is  ft  +  a  -  |  a,  or  ft  +  i  a. 

Now  the  forces  on  the  balls  are  proportional  to  the  angles  of 
twist. 

Hence2 

Force  in  position  2      /?  +  \a 
Force  in  position  1  a 

1  If  a  be   the  length  AB,  the   distance  apart  of  the  two  balls  is 
sin^a,  and  if  a  is  not  large  this  is  approximately  proportional  to  a. 

In  careful  work  however  this  'formula  should  be  used. 

2  The  formulae  may  be  put  more  accurately  thus. 

Let  a  be  the  angular  distance  between  the  balls  in  the  first  instance, 
7  in  the  second,  and  /3  the  angle  the  upper  end  of  the  wire  has  been 
turned  through ;  the  angles  of  twist  of  the  wire  to  which  the  forces  are 
proportional  are  a  and p  +  y  respectively;  the  distances  between  the  balls 
are  2a  sin  \  a  and  2a  sin  \  y. 

Thus  it  is  found  by  making  a  number  of  observations  that 

a  sin2  ^  a  =  (/3  +  7)  sin2  £7. 

Thus  if  we  call  Fl,  F2  etc.  the  forces  in  the  various  positions,  rlt  r2 
the  distances  apart,  we  see  that 

F1r1*  =  Ftr9*  = , 

or  if  F  be  the  force  at  distance  r  the  product  Fr-  is  constant  for  all 
distances. 

Hence  it  follows  that  F  is  inversely  proportional  to  r2. 


90 


ELECTRICITY 


[CH.  VI 


But  on  making  the  experiment  it  is  found  that  approxi- 
mately j3—  3Ja. 

Thus  (3  +  Ja  =»  4a,  and  the  ratio  of  the  forces  is  four  to  one, 
or  by  halving  the  distance  between  the  balls  the  force  is 
quadrupled.  If  the  distance  is  reduced  to  one-third  of  its 
original  value  the  force  is  found  to  be  increased  nine  times, 
and  so  on. 

Hence  the  force  is  inversely  proportional  to  the  square  of 
the  distance. 

As  an  example  we  might  note  that  the  angle  between 
the  balls  in  the  first  position  was  36°.  On  twisting 
the  upper  end  of  the  wire  until  the  angular  distance  was 
halved,  we  should  find  that  the  twist  required  was  about  1 26°. 
The  total  twist  on  the  wire  would  then  be  126°  +  18°  or  144° ; 
but  this  is  four  times  36°.  Hence  we  infer  that  the  force  in 
the  second  position  is  four  times  that  in  the  first. 


Fig.  52. 

60.     The  Attracted  Disc  Electrometer.     Let  AB, 

Fig.  52,  represent  a  circular  disc  supported  horizontally  from 
one    arm   of   a   balance    and    counterpoised  if   necessary    by 


59-60]  MEASUREMENT    OF    POTENTIAL  91 

weights  in  the  other  pan,  and  let  CD  be  a  second  disc  fixed 
in  a  horizontal  position  at  a  small  distance  from  AB  and 
insulated;  let  AB  be  connected  to  earth.  If  now  CD  be 
electrified  A  B  will  be  attracted  and  weights  must  be  put  into 
the  opposite  pan  to  maintain  the  beam  in  its  horizontal  position. 
There  will  be  a  relation  between  these  weights,  which  measure 
the  attraction  between  the  two  discs,  and  the  potential  to 
which  CD  has  been  electrified,  and  by  observing  the  weights 
the  potential  of  CD  can  be  calculated  from  this  relation. 

This  is  the  principle  of  the  attracted  disc  electrometer. 

If  we  knew  the  distribution  of  the  lines  of  force  between 
AB  and  CD  it  would  be  possible  to  calculate  the  attraction  on 
AB.  With  two  discs  as  described  it  would  be  difficult  to 
determine  this  distribution ;  the  lines  of  force  in  the  centre 
of  the  discs  would  run  in  straight  lines  from  one  disc  to  the 
other ;  at  the  edges  however  they  would  bulge  outwards  and 
the  calculation  of  the  attraction  would  be  troublesome  if  not 
impossible.  This  difficulty  is  avoided  by  making  the  moveable 
disc  AB  the  central  portion  of  a  very  much  larger  plate ;  the 
disc  is  separated  from  the  plate  by  a  very  narrow  aperture 
just  sufficiently  wide  to  allow  the  disc  to  move  freely  through 
it.  In  this  case  we  may  treat  the  lines  of  force  over  AB  as 
straight  lines  perpendicular  to  AB.  The  fixed  outer  portion 
of  the  disc  AB  is  known  as  the  guard-ring. 

In  the  above  we  have  supposed  the  measurement  made  by 
determining  the  weights  required  to  balance  the  electrical 
attraction,  we  may  use  the  instrument  in  a  different  manner ; 
clearly  if  we  decrease  the  distance  between  the  two  discs  we 
increase  the  attraction  and  vice  versa.  Now  let  us  suppose 
that  before  CD  is  brought  near,  the  weights  in  the  scale-pan 
are  slightly  too  heavy,  so  that  the  pointer  of  the  balance  is  a 
little  to  the  left  of  the  centre  of  the  scale.  Then  on  electri- 
fying CD  and  bringing  it  near,  we  attract  AB  and  we  can 
determine  the  position  of  CD  required  to  bring  the  pointer  to 
the  centre  of  the  scale,  into  the  sighted  position  we  may  call  it. 

Let  us  suppose  that  the  potential  of  CD  is  V  and  the 
distance  between  the  discs  is  a.  Suppose  now  that  CD  is 
electrified  to  a  different  potential  V  and  that  when  AB  is 
again  brought  into  the  sighted  position  the  distance  apart 


92  ELECTRICITY  [CH.  VI 

is  a'.  We  know  that  the  attraction  on  the  disc  AB  is  the  same 
in  these  two  positions,  for  the  weight  has  not  been  changed  ; 
but  this  attraction  depends  on  the  resultant  electrical  force 
acting  at  each  point  of  AB. 

Let  us  draw  the  equipotential  surfaces  in  the  two  cases, 
in  each  case  they  will  be  planes  parallel  to  the  disc,  and  in 
each  case  the  consecutive  surfaces  will  be  at  a  constant  dis- 
tance apart  ;  in  the  first  case,  since  there  are  V  surfaces  in  a 
distance  a  centimetres,  for  the  potential  of  CD  is  F,  of  AB  it 
is  zero,  the  distance  between  consecutive  surfaces  is  a/V.  In 
the  second  case  it  is  a/V. 

But  if  the  two  fields  of  force  be  the  same  the  resultant 
force  at  each  point  of  AB  is  the  same  in  both  cases  and  the 
charge  on  AB  is  the  same;  hence  the  attraction  on  AB  will 
be  the  same.  The  condition  for  this  will  be  that  the  distance 
between  consecutive  surfaces  should  be  the  same  in  the  two 
cases,  and  for  this  we  must  have 

a       a' 


T=a" 

Thus  we  can  compare  the  potentials  of  two  bodies  by 
comparing  the  distance  between  the  two  discs  required  to 
bring  AB  into  the  sighted  position  when  first  the  one  body 
and  then  the  second  is  connected  to  CD. 


Fig.  53. 

In  practice  it  would  be  inconvenient  to  suspend  the 
disc  from  a  balance,  and  the  instrument  usually  takes  one 
of  the  two  forms  shewn  in  Figs.  53  and  54.  In  Fig.  53 


60-61]  MEASUREMENT   OF   POTENTIAL  93 

the  disc  p  is  carried  at  one  end  of  a  long  lever.  This 
is  attached  to  a  horizontal  wire  tightly  stretched  as  at 
f  and  balanced  by  a  counterpoise.  The  torsion  of  the  wire 


J~ 

rS 

^>— 

B 

s 

c 

u 

Fig.  54. 

D 

is  so  adjusted  that  in  the  normal  position  the  disc  p  is 
slightly  above  the  guard-ring  G.  The  arm  supporting  the 
disc  carries  a  light  pointer  which  moves  in  front  of  a  vertical 
scale  which  is  viewed  by  a  lens  /.  On  the  scale  are  two  marks 
which  are  so  adjusted  that  when  the  pointer  bisects  the 
distance  between  them  the  disc  is  in  its  sighted  position ;  this 
is  easily  determined  by  the  aid  of  the  lens.  This  arrangement 
is  adopted  in  Lord  Kelvin's  portable  electrometer. 

In  the  form  shewn  in  Fig.  54  the  disc  S  is  supported  by 
three  fine  springs,  two  are  shewn  in  the  figure,  which  keep  it 
normally  slightly  above  the  guard-ring  AB  •  the  experiment 
consists  as  before  in  bringing  the  lower  disc  up  until  the  disc 
reaches  the  sighted  position,  in  which  as  illustrated  it  is  in 
the  same  plane  as  AB.  This  is  determined  by  the  aid  of  a 
lens,  if  it  be  wished  to  find  the  force  exerted  by  the  springs 
this  can  be  done  by  carefully  loading  the  disc  and  observing 
the  weights  required  to  bring  it  to  the  sighted  position.  This 
is  the  arrangement  in  Lord  Kelvin's  absolute  electrometer. 

61.      Electrostatic    Measurement    of   Potential. 

We  can  determine  the  relation  between  the  potential  of  CD 
and  the  force  of  attraction  in  the  absolute  electrometer  thus. 
If  R  is  the  resultant  force  near  a  charged  conductor,  and  cr  the 
surface  density,  then  it  follows  from  the  mathematical  theory 
that  the  pull  on  the  surface  per  unit  of  area  of  the  surface  is 
\Ra-\  Now  in  the  case  in  point  the  pull  is  in  the  same 
direction  at  each  point.  Thus  the  attraction  on  a  surface  of 
area  S  is  J#.  <r.  S. 

1  Elements  of  Electricity  and  Magnetism,  J.  J.  Thomson,  §  37. 


94  ELECTRICITY  [CH.  VI 

But  by  Coulomb's  law  R  =  4?ro-.     Thus, 

the  attraction  =  —  IPS  =  °2irv*S. 

OTT 

Moreover  since  the  force  is  the  rate  of  change  of  potential  and 
since  this  is  uniform  and  equal  to  V/a  we  have 

72 

»-£• 

1      F2 
Hence  the  attraction  is  equal  to  -^-  .  — -  .  S. 

OTT     a2 

Now  let  M  be  the  mass  added  in  the  experiment  with  the 
balance  or  the  mass  required  to  bring  the  disc  to  the  sighted 
position  in  an  experiment  with  the  springs,  then  it  is  clear  that 
the  attraction  is  equal  to  the  weight  of  the  mass  M. 


Hence  the  attraction  is  Mg. 

1 

Sir 


1    F2 
Thus  Mg=      -\-.S, 


. 

In  this  expression  a  is  measured  in  centimetres,  S  in  square 
centimetres,  M  in  grammes,  and  g  in  centimetres  per  second 
per  second,  its  value  being  very  nearly  981. 

62.  The  Quadrant  Electrometer.  In  this  instru- 
ment, shewn  diagrammatically  in  Fig.  55,  a  light  disc  of 
aluminium  known  as  the  needle  is  supported  in  a  horizontal 
position  by  a  wire,  any  force  tending  to  displace  it  from  its 
position  of  equilibrium  is  opposed  by  the  torsion  of  the  wire. 
Four  metal  quadrants  A,  £,  C,  D  each  carried  by  an  insulating 
stem  are  placed  horizontally  below  the  needle,  which  lies 
symmetrically  with  regard  to  them.  The  quadrants  A  and  D 
are  connected  together,  as  likewise  are  B  and  C.  Suppose 
now  that  the  needle  is  electrified.  If  the  four  quadrants  be 
at  the  same  potential  its  equilibrium  is  not  affected  ;  if,  how- 
ever, there  be  a  difference  of  potential  between  A  and  B,  each 
end  of  the  needle,  if  its  own  potential  be  positive,  is  repelled 


(51-62] 


MEASUREMENT   OF    POTENTIAL 


95 


from  the  quadrants  at  high  potential  towards  those  at  low, 
the  needle  therefore  is  twisted  until  this  repulsion  is  balanced 
by  the  torsion  of  the  wire,  and  the  angle  through  which  it  is 
twisted  can  be  shewn  to  be  proportional  to  the  difference 
of  potential  between  the  quadrants. 


Fig.  55. 

In  practice  the  quadrants  are  doubled,  a  set  being  placed 
over  as  well  as  below  the  needle,  the  corresponding  quadrants 
being  connected  together,  so  that  the  needle  is  practically 
entirely  surrounded  and  hangs  in  a  kind  of  hollow  box. 

To  measure  the  deflexion  of  the  needle  a  mirror  is  attached 
to  it,  and  reflects  a  spot  of  light  on  to  a  scale.  In  order  to 
use  the  instrument  the  four  quadrants  are  connected  together, 
and  adjusted  until  the  needle  hangs  symmetrically,  the  con- 
nexion is  then  removed,  and  the  position  of  the  spot  is 
observed ;  one  pair  of  quadrants  is  then  usually  put  to  earth, 
the  other  is  connected  to  the  body  whose  potential  is  required, 
and  the  deflexion  of  the  spot  is  measured.  This  is  approxi- 
mately proportional  to  the  potential. 

If  instead  of  connecting  one  pair  of  quadrants  to  the 
earth  we  connect  it  to  a  body  at  potential  V  and  if  V  be  the 
potential  of  the  other  pair,  V0  that  of  the  needle, — in  practice 


96 


ELECTRICITY 


[CH.  VI 


F0   is   large    compared    to   V  or    V,  —  then    the   deflexion    is 
approximately  proportional1  to  F0  (V  —  V). 

Thus  it  is  necessary  that  the  potential  of  the  needle  should 
remain  the  same  throughout  the  observations;  now  the  capacity 
of  the  needle  is  small,  and  therefore  a  small  leakage  may  make 
a  considerable  change  in  its  potential.  To  overcome  this  a 
piece  of  platinum  wire  hangs  from  the  needle  into  a  glass  vessel 
of  strong  sulphuric  acid  placed  beneath  the  quadrants.  The 
vessel  is  coated  outside  with  tinfoil,  which  is  earthed  by 
contact  with  the  case  of  the  instrument,  and  thus  forms  a 
Leyden  jar  of  large  capacity  with  the  sulphuric  acid  for  the 
inner  coating.  By  this  means  the  rate  of  fall  of  potential 
of  the  needle  is  checked. 

The   instrument    is    covered   with    a  glass  case,   and   the 
sulphuric  acid  serves  in  addition 
to  keep  the  air  within  the  case 
free  from  moisture. 

Fig.  56  shews  a  simple  form 
of  quadrant  electrometer. 

EXPERIMENT  15.  To  shew  that 
the  potential  difference  between  the 
quadrants  of  a  quadrant  electro- 
meter is  proportional  to  the  de- 
flexion. 

Take  three  or  four  condensers 
of  equal  capacity,  —  some  Leyden 
jars  made  of  the  same  kind  of 
glass,  of  the  same  size  and  thick- 
ness will  be  convenient,  —  insulate 
all  but  the  last  and  connect  them 
in  "cascade2"  as  shewn  in  Fig. 
57,  where  we  suppose  there  are 
four  condensers.  Fig.  56. 

1  The  more  complete  formula  is  that  the  deflexion 


where  A;  is  a  constant  depending  on  the  instrument.  From  this  the 
result  in  the  text  follows  by  supposing  V  and  V  small  compared  with  F0  . 
2  See  Section  45.  Instead  of  the  Leyden  jars  a  number  of  brass 
plates  of  equal  area  placed  in  a  pile  at  equal  distances  apart  and 
insulated  by  small  pieces  of  ebonite  may  be  used. 


62-63]  MEASUREMENT  OF   POTENTIAL  97 

Let  Alt  A^  As,  A 4  be  the  successive  inner  coatings,  B1}  B2,  B3, 
B4  the  outer  coatings ;  A2  is  connected  to  B1,  A3  to  B2)  and  so  on. 
Charge  A1  to  a  potential  F;  the  potential  of  B^  is  zero,  for 


Fig.  57. 

it  is  earth-connected,  and  the  fall  of  potential  from  V  to 
nothing  will  be  equally  divided  among  the  four  condensers. 
This  can  be  shewn  by  experiment1,  for  if  we  connect  the  two 
coatings  of  each  condenser  in  turn  to  a  quadrant  electrometer 
we  find  we  obtain  the  same  deflexion. 

Thus  there  is  a  difference  of  potential  F/4  between  the 
coatings  of  each  condenser. 

Thus  between  Al  and  Bl  the  potential  difference  is  F/4. 
Between  Al  and  B%  it  is  2  F/4. 
^and  J53it  is  3  F/4. 
A,  and  B^  it  is  4  F/4. 

Now  connect  Al  to  one  pair  of  quadrants  and  connect  in 
turn  to  the  other  pair  Slt  Bz,  Bs,  B4  noting  the  deflexion  in 
each  case.  They  will  be  found  to  be  respectively  proportional  to 
1,  2,  3,  and  4.  Hence  the  deflexions  observed  are  proportional 
to  the  potential  differences  between  the  quadrants.  Thus  we 
may  use  the  quadrant  electrometer  to  compare  differences  of 
potential  by  comparing  the  deflexions  produced. 

Moreover,  if  we  can  determine  in  any  way  the  potential 
difference  which  produces  a  given  deflexion  we  can  find  the 
constant  of  the  instrument  and  so  use  it  to  measure  any  other 
difference  of  potential. 

63.    Electrostatic  and  Multicellular  Voltmeters. 

The  quadrant  electrometer  was  invented  by  Lord  Kelvin,  who 
has  given  it  various  forms.  In  the  electrostatic  voltmeter2 

1  This  follows  at  once  from  the  theory  given  in  Section  45,   if  we 
suppose  all  the  condensers  to  have  equal  capacity. 

2  A  volt  is  the  name  given  to  the  unit  difference  of  potential,  and  a 
voltmeter  is  an  instrument  for  measuring  volts. 

G.  E.  ^ 


98 


ELECTRICITY 


[CH.  VI 


(Fig.  58)  two  opposite  pairs  of  quadrants  are  removed,  the 
axis  round  which  the  needle  turns  is  horizontal ;  the  lower 
end  of  the  needle  carries  an  adjustable  weight,  and  the  needle 


Fig.  58. 

is  connected  to  the  Quadrants.  When  the  instrument  is 
electrified  the  needle  and  quadrants  are  raised  to  the  same 
potential  and  the  needle  is  repelled  from  the  quadrants.  The 
motion  of  the  needle  is  indicated  by  a  pointer  which  moves 
over  a  graduated  scale.  In  this  case  the  force  on  the  needle 
is  proportional  to  the  square  of  the  potential ;  this  follows  at 
once  if  we  put  V0-  V,  V  =  0  in  the  formula  in  the  note  to 
Section  62,  which  then  gives  us  the  deflexion  as  equal  to  J&F2. 

We  can  also  use  the  electrostatic  voltmeter  to  measure  the 
difference  of  potential  between  two  bodies  by  connecting  one 
body  to  the  needle,  the  other  to  the  quadrants. 

In  the  multicellular  voltmeter  the  sensitiveness  of  the 
instrument  is  increased  by  attaching  to  the  same  vertical 
wire  a  number  of  needles  each  of  which  hangs  in  a  horizontal 
position.  These  are  attracted  by  a  series  of  quadrant-shaped 


63-64] 


MEASUREMENT  OF   POTENTIAL 


99 


discs  placed  between  the  needles  as  shewn  in  Fig.  59.  In 
this  instrument  also,  which  is  used  for  measuring  high  potential 
differences,  it  is  usual  to  connect  one  of  the  two  points  between 
which  the  potential  is  required  to  the  needle,  while  the  other 
is  joined  to  the  quadrants.  a. 

The  deflexion  is  then  proportional  to  the  square  of  the 
difference  of  potential. 

64.  Potential  at  a  point 
in  the  air.  It  is  clear  that  an 
insulated  conductor  need  not  be 
at  the  same  potential  as  the  air 
round  it ;  if  however  there  be  a 
difference  of  potential  between  a 
conductor  and  the  air,  and  we 
can  arrange  that  a  stream  of 
small  conducting  particles  shall 
leave  the  conductor,  since  posi- 
tive electricity  tends  to  pass  from 
places  of  high  to  places  of  low 
potential,  the  conductor,  if  at  a 
higher  potential  than  the  air,  will 
discharge  positive  electricity  with 
the  stream  of  particles,  and  con- 
versely if  it  is  at  a  lower  potential 
it  will  discharge  negative  electri- 
city ;  in  either  case  the  effect 
will  be  to  bring  the  conductor  to  the  potential  of  the  air 
in  its  neighbourhood.  If  then  the  conductor  be  connected  to 
an  electrometer  the  readings  of  the  instrument  will  measure 
the  potential  of  the  air  at  the  point  where  the  stream  of 
particles  breaks  up. 

In  some  cases  the  stream  of  particles  is  a  fine  jet  of  water 
issuing  from  a  long  nozzle  attached  to  an  insulated  vessel,  the 
vessel  is  connected  with  an  electrometer,  which  thus  measures 
the  potential  of  the  air ;  if  it  is  wished  to  obtain  a  continuous 
record  of  the  changes,  the  quadrant  electrometer  is  employed, 
and  the  spot  of  light  from  the  mirror  is  focussed  on  to  a  sheet 
of  sensitive  paper  wound  on  a  drum,  which  is  made  to  move 
by  clockwork. 

7—2 


Fig.  59. 


100 


ELECTRICITY 


[CH.  VI 


The  motion  of  the  spot  is  parallel  to  the  axis  of  the  drum, 
at  right  angles  therefore  to  the  motion  of  the  paper. 

The  trace  of  the  spot  when  developed  forms  a  continuous 
curve,  which  gives  the  variations  of  the  potential.  Such  a 
curve  obtained  at  the  National  Physical  Laboratory  is  shewn 
in  Fig.  60. 


Fig.  60. 

In  another  arrangement  due  also  to  Lord  Kelvin  the 
necessary  stream  of  particles  is 
afforded  by  the  smoke  from  a  slow- 
burning  match  :  the  match  is  held 
in  an  insulating  handle  and  con- 
nected to  the  electrometer,  and  the 
potential  measured  by  reading  the 
instrument.  The  portable  electro- 
meter (Fig.  61)  is  usually  employed 
for  this  purpose. 

Very  large  variations  of  poten- 
tial are  observed,  during  a  day  the 
changes  at  a  given  point  may 

amount  to  500  or  even  1000  volts. 

Fig.  61. 


EXAMPLES   ON   ELECTROSTATICS  101 


EXAMPLES  ON  ELECTROSTATICS. 

1.  A  rod  is  brought  near  to  a  magnetic  needle,  and  the  latter  is 
thereby  deflected.     By  what  further  experiment  would  you  determine 
whether  the  observed  action  is  magnetic  or  electrical? 

2.  A  small  positively  charged  sphere  is  placed -near  the  end  of  a 
small  insulated  uncharged  cylinder,  and  both  are  placed  near  the  centre 
of  a  large  spherical  metallic  shell  connected  with  the  earth.     Describe  in 
general  terms  the  distribution  of  the  charges  on  the  bodies,  and  sketch 
the  lines  of  force  of  the  system. 

3.  The  end  of  a  wire  connected  to  a  gold-leaf  electroscope  is  put 
through  a  hole  in  the  side  of  a  hollow  charged  conductor.     Describe 
and   explain  what  happens  when   (1)   it   is   held   there   without  being 
allowed   to   touch   the  conductor,   (2)  it  is  withdrawn,  (3)  it  is  again 
inserted   and   allowed   to  touch  the  inside  of  the  conductor,  (4)  it  is 
withdrawn,   (5)  it  is  made  to  touch  the  outside  of  the  conductor. 

4.  A  gold-leaf  electroscope  is  put  inside  a  tin  can,  which  is  hung  up 
by  silk  cords  so  as  to  be  insulated.     On  holding  a  strongly  electrified 
glass  rod  below  the  can  no  divergence  of  the  leaves  takes  place ;  but  on 
touching  the  cap  of  the  electroscope  with  the  fingers  (without  touching 
the  can)  the  leaves  diverge.     Explain  these  results. 

5.  Two  exactly  similar  gold-leaf  electroscopes  have  their  caps  con- 
nected by  a  wire  and  a  positively  charged  body  is  brought  near  the  cap 
of  one  of  them.     Compare  and  explain  their  indications. 

If  the  wire  be  now  removed  by  means  of  an  insulating -handle  and 
then  the  charged  body  also  removed,  what  effects  will  be  observed  in  the 
electroscopes  ? 

6.  A  hollow  cylinder  is  charged  positively  and  insulated.     An  in- 
sulated metal  funnel  is  placed  with  its  nozzle  well  within  the  cylinder 
and  water  is  allowed  to  drop  through  the  funnel  into  an  insulated  vessel 
placed  below  the  cylinder.     Shew  that  the  vessel  will  become  negatively 
electrified. 

7.  A  large  box  is  coated  with  tinfoil  and  insulated.    A  wire  insulated 
from  the  walls  of  the  box  and  connected  to  the  earth  passes  to  the  inside. 
How  can  a  man  inside  the  box,  if  he  has   the   necessary  apparatus, 
determine  whether  the  box  be  charged  or  not? 

8.  Would  an  electrophorus  work  better  when  the  cake  of  pitch  or 
resin  is  placed  on  a  conducting  plate  or  when  it  is  in  contact  with 
non-conductors  only? 

9.  How  could  you  give  to  a  hollow  vessel  T>  double  the  charge  of  a 
small  conductor  A  without  discharging  A  ? 

10.  If  a  charged  ebonite  rod  be  placed  in  contact  with  the  knob  of 
an   electroscope  the  leaves  diverge  and  on  its  removal  they  partially 
collapse.     Why  is  this? 


102  ELECTRICITY  [CH.  VI 

11.  If  an  electroscope  be  charged  and  a  body  with  a  big  charge  be 
brought  near  it,  state  and  explain  what  will  be  the  indications  of  the 
electroscope  according  as  the  charges  are  of  the  same  or  of  opposite  sign  ? 

12.  If  a  small  spherical  conductor  with  a  strong  positive  charge  is 
gradually  brought  near   to   a  large   spherical  conductor  with   a  weak 
positive  charge  repulsion  followed  by  attraction  is  experienced.    Ex- 
plain this. 

13.  A  hollow  insulated  conductor  is  electrified  positively.    *A  small 
insulated  conducting  sphere,  connected  by  a  fine  wire  with  a  gold-leaf 
electroscope  which  is  negatively  electrified,  is  placed  (a)  near  the  outside ; 
(b)  touching  the  outside;  (c)  inside;  (d)  touching  the  inside  of  the  hollow 
conductor.     Describe  and  explain  in  each  case  the  result,  on  the  electro- 
scope, of  the  experiment. 

14.  A  ball  held  by  a  damp  silk  thread  is  introduced  inside  a  charged 
hollow  conductor,  and  made  to  touch  the  side.     It  is  brought  out  and 
presented    to    an    electroscope    which    indicates    a    charge.      Explain 
this  fact. 

15.  Two  equal  small  spheres  are  charged  when  in  contact  and  then 
placed  at  a  distance  of  1  metre  apart.     They  are  found  to  repel  each 
other  with  a  force  of  900  dynes.     Find  the  charge  on  either. 

16.  What  charge  shared  between  two  equal  spheres  5  cm.  apart  will 
cause  them  to  repel  each  other  with  a  force  of  81  dynes? 

17.  Two  small  equal  metal  spheres  are  placed  5  cm.  apart.     What 
is  the  force  between  them  if  one  has  a  charge  of  +  5  units  and  the  other 
- 10  units  ?     What  does  the  force  become  if  they  are  connected  for  a 
moment  by  a  wire?     Would  there  be  any  force  if  one  of  them  were 
connected  with  the  earth  and  the  other  charged? 

18.  Two  small  spheres  a  and  b  charged  with  the  same  quantity  of 
positive  electricity  are  placed  at  a  distance  of  1  metre.     Where  should  a 
sphere  c  holding  twice  the  amount  of  electricity  be  placed  so  that  the 
electrical  forces  on  6  may  be  in   equilibrium  (1)  when  c  is   charged 
positively,  (2)  when  c  is  charged  negatively? 

19.  Sketch  the  lines  of  force  and  the  equipotential  surfaces  due  to 
two  small  spheres  placed  1  metre  apart  (a)  when  charged  with  quantities 
+  400  and  + 100  units  respectively,  (b)  when  charged  with  quantities  +  400 
and  -  100  respectively ;  and  find  in  each  case  the  position  of  the  point  at 
which  the  force  is  zero. 

20.  Two  small  pith  balls,  each  weighing  one  gram,  are  suspended 
from  the  same  point  by  silk  fibres  each  12  cm.  long,  and  of  negligible 
mass.     They  are   then   equally  electrified  so  that   each   string  makes 
an  angle  of  45°  with  the  vertical.     What  charge  must  each  ball  possess  ? 

21.  It  is  required  to  determine  which  of  two  electrical  conductors 
has  the  larger  capacity.     Explain  how  you  would  decide  the  question 
if  you  were  provided  with   a   sensitive   gold-leaf  electroscope  and  an 
electrophorus. 


EXAMPLES   ON   ELECTROSTATICS  103 

22.  The  knob  of  an  electroscope  is  connected  with  an  insulated 
metal  plate :  this  is  then  charged  with  electricity :  a  metal  plate,  held  in 
the  hand,  is  brought  up  opposite  the  insulated  plate;  state  and  explain 
what  will  be  the  indications  of  the  electroscope. 

23.  Three  Leyden  jars  whose  capacities  are  C^,  <7a,  (7,  are  insulated, 
except  that  the  outer  coating  of  the  third  jar  is  earthed;   the  outer 
coating  of  the  first  is  connected  to  the  inner  coating  of  the  second, 
and  the   outer    coating    of  the    second    to    the    inner    coating   of  the 
third.     A  charge  Q  is  given  to  the  inner  coating  of  the  first.     What 
are  the  charges  on  the  inner  coatings  of  the  other  two,  and  what  is  the 
difference  of  potential  between  the  inner  coating  of  the  first  and  the  outer 
coating  of  the  third  ? 

24.  One  plate  of  a  plate   condenser  is   connected  to   a  gold-leaf 
electroscope  and  the  following  operations  are  performed:    (1)  the  other 
plate  is  charged  positively;    (2)  the  electroscope  is  momentarily  put  to 
earth  and  then  again  insulated;    (3)  the  distance  between  the  plates  is 
increased;  (4)  a  plate  of  glass  is  inserted  between  the  plates.     Describe 
and  explain  the  effect  on  the  electroscope  in  each  case. 

25.  Two  metal  spheres,  one  six  times  the  diameter  of  the  other,  are 
connected  by  a  long  thin  wire  and  electrified.     Compare  their  electric 
charges,  potentials,  densities  and  energies. 

26.  Find  the  work  spent  in  giving  a  sphere  of  10  cm.  radius  a  charge 
of  50  units. 

27.  The  capacity  of  a  conductor  is  20  c.  o.  s.  units.     What  must  be 
its  charge  in  order  that  its  energy  may  be  1000  ergs? 

28.  How  is  the  electrical  energy  of  a  charged  air  condenser  affected 
if  a  cake  of  sulphur  is  introduced  between  the  plates  ? 

29.  A.  body  of  capacity  C  is  charged  to  a  potential  V.     What  change 
in    energy   occurs   when  its   charge   is   shared   with    another    body  of 
capacity  C"? 

30.  An  insulated  charged  sphere,  3  cm.  in  radius,  shares  its  charge 
with  another,  6  cm.  in  radius;  what  is  the  relation  between  the  energy 
of  the  charged  body  before  and  after  it  has  shared  its  charge? 

31.  The  opposite  plates  of  an  air- condenser  are  connected  with  a 
quadrant  electrometer,  which  shews  a  small  deflexion  6  when  the  con- 
denser is  charged  by  means  of  a  DanielPs  cell.    What  will  be  the  deflexion 
if  the  air  be  replaced  by  a  slab  of  paraffin  (specific  inductive  capacity  =  2), 
(i)  before,  and  (ii)  after  the  cell  is  disconnected  from  the  condenser  ? 

32.  Three  insulated  conducting  spheres,  whose  radii  are  respectively 
2,  3  and  4  cm. ,  are  so  charged  that  their  respective  potentials  are  6,  4  and 
3  units  of  potential.     They  are  then  connected  by  wires  of  negligible 
capacity.     Find  the  total  quantity  of  charge,  the  total  capacity,  and  the 
final  common  potential  of  the  system. 


104  ELECTRICITY  [CH.  VI 

33.  An  insulated  cylindrical  sheet  of  tinfoil  is  electrified  so  that  two 
pith-balls,  suspended  from  the  foil  by  cotton  threads,  diverge  widely. 
The  foil  is  then  unrolled  (still  remaining  insulated)  and  it  is  found  that 
the  angle  of  divergence  is  diminished.     Explain  this  result. 

34.  An  air-condenser  is  charged  and  connected  to  an  electrometer ; 
the  distance  between  the  plates  of  a  condenser  is  measured ;   a  plate 
of  ebonite  of  known  thickness  is  inserted  between  the  plates  and  their 
distance  is  adjusted  so  that  the  electrometer  deflexion  does  not  change. 
Shew  how  to  determine  hence  the   specific  inductive   capacity  of  the 
ebonite. 

35.  Find  the  capacity  of  a  condenser  composed  of  a  pair  of  circular 
plates  18  cm.  diameter  and  1  mm.  apart,  with  air  as  the  dielectric. 

36.  Find  the  capacity  of  a  spherical  condenser  the  radii  of  whose 
surfaces  are  15-9  and  16-1  cm.  respectively.     What  would  be  the  effect 
of  replacing  the  air  between  the  spheres  with  turpentine? 

37.  A  condenser  is   composed  of  two  square   plates  each   10  cm. 
in   side;   the   plates  are  1  mm.  apart.     It  is  found  that  500  ergs   are 
needed  to  charge  it  to  a  constant   difference   of  potential.     Find   the 
difference  of  potential  between   the  plates   and  the   charge   on  either 
plate. 

38.  A  Leyden  jar  is  charged  to  potential  1200.     It  is  then  connected 
to  another  jar  with  twice  the  area  of  coating  but  of  the  same  thickness 
and  previously  uncharged.   What  is  the  common  potential  ?  If  the  second 
jar  had  been   previously  charged  to  potential  600  what  would  be  the 
common  potential? 

39.  The  diameter  of  a  Leyden  jar  is  12cm.,  the  height  of  the  metal 
coating  15  cm.,  the  thickness  of  the  glass  2  mm.,  and  its  specific  inductive 
capacity  3-1.    Calculate  the  electrostatic  capacity,  and  find  the  work  done 
in  charging  it  to  a  potential  V. 

40.  Calculate  the  total  heat  developed  by  the  discharge  of  a  con- 
denser consisting  of  two  concentric  spheres  separated  by  paraffin  when 
the  charge  on  one  sphere  is  100  electrostatic  units.     The  radii  of  the 
spheres  are  10  and  12  cm.  and  the  dielectric  constant  of  paraffin  is  2. 

41.  An  attracted  disc  electrometer  is  immersed  in  an  insulating  oil 
with  specific  inductive  capacity  2.     The  area  of  the  attracted  disc  is 
50  sq.  cm.,  and  its  distance  from  the  fixed  plate  is  0'5  cm.     If  the  electric 
pull  on  the  disc  is  500  dynes,  find  its  difference  of  potential  from  that  of 
the  fixed  plate. 


CHAPTER   VII. 

MAGNETIC  ATTRACTION  AND   REPULSION. 

65.  Natural  Magnets.  It  was  known  to  the 
ancients  that  certain  black  stones — iron  ores,  which  were 
found  commonly  at  Magnesia  in  Asia  Minor — possessed  the 
power  of  attracting  to  themselves  small  pieces  of  iron.  Such 
stones  were  called  magnets.  Many  centuries  later  it  was 
found  that  a  magnet  when  suspended  freely  by  a  thread 
tended  always  to  set  itself  in  one  definite  position1.  One  end 
was  observed  always  to  point  north,  the  other  south.  Hence 
the  magnet  acquired  the  name  of  lodestone  or  leading  stone. 
The  only  force  we  know  of  acting  between  the  earth  and 
most  bodies  is  that  of  gravitation  ;  in  the  case  of  a  magnet 
an  additional  magnetic  force  comes  into  play  and  the  position 
the  magnet  assumes  depends  in  part  on  this  force. 

These  natural  magnets  are  composed  of  an  oxide  of  iron, 
having  the  chemical  composition  Fe304,  and  are  found  in  many 
parts  of  the  world  besides  Magnesia. 

After  a  time  it  was  discovered  that  these  magnetic  pro- 
perties could  be  communicated  to  a  piece  of  iron  which 
was  stroked  by  a  natural  magnet,  and  still  later  it  was 
shewn  that  hardened  steel  when  stroked  retained  its  magnetic 
properties  more  permanently  than  soft  iron. 

Dr/  Gilbert,  who  in  1600  published  a  book,  de  Magnete, 
about  magnetism,  added  much  to  our  knowledge  by  his 
discoveries.  He  shewed  for  example  that  if  a  bar  of  steel 

1  It  is  said  that  the  Chinese  were  acquainted  with  this  property  at  a 
very  early  date. 


106  MAGNETISM  [CH.  VII 

whose  length  is  considerable  in  comparison  with  its  width  be 
magnetised,  the  magnetic  attraction  is  exerted  most  markedly 
by  the  ends  of  the  bar,  and  hence  he  introduced  the  idea  of 
the  magnetic  poles  of  a  magnet,  two  points  towards  which  he 
supposed  the  attractive  forces  to  act.  The  line  joining  these 
two  poles  he  called  the  axis  of  the  magnet. 

Moreover  he  proved  that  this  magnetic  attraction  was 
exerted  across  almost  all  kinds  of  bodies  practically  unchanged ; 
a  sheet  of  wood  or  brass  or  lead  may  be  inserted  between  the 
magnet  and  the  soft  iron  without  modifying  the  effect.  The 
magnet  may  be  enclosed  in  a  vessel  or  sealed  up  in  a  glass 
tube  containing  air  or  any  gas  or  exhausted,  as  the  case  may 
be,  the  force  remains  unaltered.  A  screen  of  iron  however 
will  affect  the  action  of  the  force.  A  magnet  inside  a  very 
thick  hollow  shell  of  soft  iron  exerts  no  sensible  force  outside 
the  shell  and  is  unaffected  by  external  magnets. 

It  is  known  now  that  an  actual  magnet  does  not  possess  "poles," 
two  centres,  that  is,  of  attraction  ;  the  axis  of  a  magnet  however  is  a  term 
which  we  shall  use  continually,  and  of  which  we  can  give  a  definition 
quite  independent  of  the  idea  of  poles  ;  at  the  same  time  in  dealing  with 
long,  thin  magnets  it  is  convenient  to  use  the  term  pole  as  indicating  the 
end  of  the  magnet,  and  as  we  shall  see  the  effects  of  the  magnet  can  be 
calculated  approximately  as  though  the  poles  were  real  centres  of 
attraction. 

66.  Artificial  Magnets.  A  steel  magnet  may  take  many 
forms ;  a  long  rectangular  bar,  such  as  is  shewn  in  Fig.  62,  is 


Fig.  62.  Fig.  63. 


65-66]        MAGNETIC   ATTRACTION   AND  REPULSION          10*7 

known  as  a  bar  magnet ;  sometimes  the  bar  is  bent  into  the 
form  of  a  horse-shoe  (Fig.  63).  In  a  compass  needle  (Fig.  64) 
a  thin  strip  of  steel  in  the  form  of  a  very  elongated  lozenge  is 

r\ 


Fig.  64.  Fig.  65. 

employed.  This  carries  at  its  centre  a  small  cup  of  glass  or 
agate  by  which  it  can  be  supported  on  a  sharp  point.  For 
many  of  our  experiments  a  thin  knitting  needle  will  be  found 
useful ;  this  can  if  necessary  be  suspended  in  a  small  stirrup 
of  brass  or  copper  wire  by  means  of  a  silk  fibre  (Fig.  65). 


N 

Fig.  65  a. 

A  spherical  ended  magnet  (Fig.   65  a)  as  constructed  by 
Robeson  may  conveniently  be  made  out  of  two  J  inch  steel 


108 


MAGNETISM 


[CH.  VII 


bicycle  balls  connected  by  a  piece  of  knitting  needle  six  inches 
in  length ;  in  such  a  magnet  the  poles  coincide  very  nearly 
with  the  centres  of  the  balls. 

xi  67.  Magnetic  Attractions.  The  attraction  of  a 
magnet  for  iron  may  be  illustrated  in  many  ways.  If  a 
magnet  be  dipped  into  a  vessel  of  iron  filings  the  filings  adhere 
to  the  magnet,  and  in  general  it  will  be  observed  that  the 
filings  are  most  ^thickly  distributed  near  the  ends  of  the 
magnet. 

Take  a  compass  needle  or,  if 

magnet  in  its  stirrup  by  a  silk  fibre.  Note  the  directioj>xm 
which  it  places  itself,  and  observe  that  after  it  is  disturbed  it 
comes  to  rest  again  in  the  same  position  as  before,  one  end, 
which  we  will  call  the  north  pole,  points  to  the  north,  the 
other,  the  south  pole,  points  to  the  south.  Bring  near  to 
either  pole  of  the  magnet  various  substances  in  succession, 
such  as  rods  of  glass,  wood,  copper,  brass,  lead,  etc.,  and 
observe  that  none  of  these  appreciably  disturb  the  magnet. 
Now  bring  a  piece  of  iron  near  either  pole ;  there  is  attraction 
between  the  iron  and  the  magnet,  and  the  magnet  is  deflected. 
The  iron  is  said  to  be  a  magnetic  substance;  most  of  the 
others  were  called  by  Gilbert  non-magnetic ;  more  careful 
observation  however  shewed  that  some  of  the  above-mentioned 
bodies  are  slightly  repelled  by  the  magnet.  Such  bodies  have 
been  called  by  Faraday  dia-magnetic,  while  iron  and  the  other 
magnetic  bodies  which  are  attracted  were  named  para-mag- 
netic. Other  substances  which  are  attracted  by  the  magnet, 
and  are  therefore  para-magnetic,  are  nickel  and  cobalt,  but 
the  attraction  exerted  on  these  bodies  is  very  much  less  than 
in  the  case  of  iron.  In  fact  for  our  purposes  we  may  look  upon 

steel — as  magnetic,  and  other  subs^ances__as_— 
non-magnetic.  /  We  must  carefully  distinguish  between  a 
magnet  which  sets  in  a  definite  position  and  attracts  iron 
fiUn^Xalid^ar-Timgiietic  substance,  such  as  iron,  j  which  is 
"aTfracted  by  a  magnetjfbut  which  in  its  normal  - condition 
does  not  attract  other  magnetic  material.  A 


substance 


stance   can-  15e  magnetised";    a  magnet_is  a  pieceof  such  a 
been   magn0tioccl~      We   have~  tacitly 


_ 

assumed  above  that  the  iron  rod  is  not  itself  a  magnet  ;   it 


66-67]        MAGNETIC   ATTRACTION    AND   REPULSION 


109 


cj 

\     mag 


remains  to  describe  what  happens  when  we  bring  a  second 
magnet  near  the  first. 

EXPERIMENT  16.  To  examine  the  forces  between  two 
magnetic  poles. 

Take  two  compass  needles  and  support  them  at  some 
distance  apart. 

Notice  that  the  needles  point  in  the  same  direction, 
approximately  north  and  south,  and  mark  the  north  or  north- 
pointing  end  of  each. 

Dismount  one  needle  and  bring  each  end  in  turn  to  the 

'  two  ends  of  the  second  needle,  observing  the  results.     It  will 

be  found  that  when  the  two  like  ends  are  close  together  there 

is    repulsion,   when  the  two  unlike  ends   are    brought   near 

there  is  attraction.     The  results  may  be  summed  up  thus : 


Effect  on  Compass  Needle 

North  Pole 

South  Pole 

North  Pole  of  dismounted  needle 
South  Pole  of  dismounted  needle 

Eepulsive 
Attractive 

Attractive 
Repulsive 

Hence  we  arrive  at  the  result  that  there  is  repulsion 
between  two  like,  attraction  between  two  unlike  magnetic 
poles. 

We  can  vary  these  experiments  by  putting  plates  of 
various  materials  between  the  magnetic  poles.  We  shall 
find  that  unless  the  material  interposed  contains  iron1  the 
forces  between  the  poles  are  the  same  as  previously. 

Thus  we  see  that  on  a  given  magnetic  pole — the  end  of 
one  magnet — opposite  effects  are  produced  by  the  two  ends  of 
a  second  magnet.  We  may  compare  this  with  the  opposite 

1  There  are  some  few  other  substances  besides  iron,  e.g.  nickel  and 
cobalt,  which  will  modify  the  force  very  slightly,  but  so  slightly  that 
much  more  delicate  experiments  than  those  described  are  required  for  its 
detection. 


f: 


110  MAGNETISM  [CH.  VII 

effects  produced  on  a  positively  electrified  body  by  two  con- 
ductors, one  charged  positively,  the  other  negatively,  and  may 
ascribe  opposite  qualities — positive  and  negative  respectively — 
to  the  two  ends  of  a  magnet.  The  north-pointing  end  is 
called  the  positive  end;  it  is  described  as  being  positively 
magnetised,  or  as  possessing  a  positive  charge  of  magnetism ; 
it  is  called  a  positive  pole ;  the  south-pointing  rod  is  negatively 
magnetised,  or  possesses  a  negative  charge;  it  is  a  negative 
pole.  As  in  the  case  of  electricity,  the  choice  of  sign  is 
conventional. 

When  one  magnet  is  brought  near  a  second  the  result- 
ing action  is  complex ;  in  the  first  place  the  magnet  does 
not  strictly  possess  poles,  we  cannot  regard  one  point  as 
a  centre  of  attraction,  another  as  a  centre  of  repulsion,  and 
deduce  all  the  forces  from  the  action  of  these  two  centres ; 
however  with  a  long,  thin  magnet  we  may  very  approximately 
make  this  assumption,  but  even  in  this  case  we  have  four 
forces  to  consider. 

For  let  N8,  N'S'  (Fig.  66)  represent  the  two  magnets, 
then  there  are  forces  of  repulsion  along  NN'  and  SS'  re- 
spectively, and  of  attraction  along  NS'  and  WS.  If  however 


Fig.  66. 

'  is  small  and  the  other  distances  considerable,  the  resulting 
action  will  be  mainly  due  to  the  repulsion  along  N1V'. 

Again,  although  it  is  convenient  to  speak  of  a  magnetic  pole  and  of 
the  force  exerted  by  such  a  pole  we  must  remember  that  we  can  never 
obtain  a  single  pole.  Any  magnet  has  always  two  poles  ;  if  we  have  an 
electrical  conductor  positively  electrified  at  one  end,  negatively  at  the 


67-69]        MAGNETIC    ATTRACTION    AND   REPULSION          111 

other,  and  divide  it  into  two  portions  we  can  separate  the  positive  and 
the  negative  electrification,  we  cannot  do  this  with  a  magnet ;  if  we  take 
a  magnet  NS  and  cut  it  in  two  in  the  middle,  each  of  the  two  halves 
remains  a  magnet  with  a  north  and  a  south  pole. 

68.  Magnetic    Induction.      A   magnetic    substance 
may  be  temporarily  magnetised  by  the  presence  of  a  magnet. 
Take  a  rod  of  soft  iron  and  dip  it  into  a  vessel  of  iron  filings, 
on  removing  it  few1  if  any  of  the  filings  adhere  to  the  iron. 
Repeat  the  experiment,  holding  one  end  of  a  powerful  bar 
magnet  near  to  the  upper  end   of  the  iron  rod.     The  iron 
filings  adhere  plentifully  to  the  rod,  but  many  drop  off  when 
the  magnet  is  removed.     The  rod  has  been  magnetised  by 
induction  ;  the  effect  is  very  much  increased  if  the  magnet  be 
allowed  to  touch  the  iron  rod. 

Thus  a  whole  string  of  iron  tacks  can  hang  suspended,  as 
shewn  in  Fig.  67,  from  one  end  of  a  magnet ;  if  the 
uppermost  tack  be  separated  from  the  magnet  the 
chain  falls  to  pieces. 

Again,  when  a  rod  is  magnetised  by  induction 
the  end  nearest  the  inducing  pole  is  of  opposite 
sign  to  that  pole.  To  shew  this  bring  one  end  of  a 
rod  of  soft  iron  near  the  north  end  of  a  compass 
needle,  it  is  attracted  to  the  iron,  then  bring  the 
north  pole  of  a  strong  magnet  near  the  further 
end  of  the  iron,  the  compass  needle  is  now  repelled ; 
the  end  of  the  iron  rod  which  originally  attracted 
the  north  end  of  the  compass  has  been  positively 
magnetised  by  induction  and  now  repels  it.  This 
enables  us  to  account  in  some  measure  for  the 
attraction  between  the  magnet  and  the  iron.  For 
let  NS  (Fig.  68)  be  the  magnet  and  AB  a  piece  of 
iron,  the  end  A  being  near  to  N  and  B  at  some  Fig.  67. 
distance.  The  rod  AB  is  magnetised  by  induction, 
A  becoming  a  south  end,  B  a  north.  The  south  end  A  is 
then  attracted  by  the  north  end  N  of  the  original  magnet. 

69.  Magnetic  Field.     Thus  a  magnet  exerts  magnetic 
force  011  any  other  magnetic  body  in  its  neighbourhood  ;   the 

1  If  the  iron  happens  to  have  been  recently  magnetised  a  few  of  the 
filings  may  adhere  io  it. 


112  MAGNETISM  [CH.  VII 

space  round  the  magnet  throughout  which  this  force  is  exerted 
is  called  its  magnetic  field,  and  at  any  point  of  the  field  there 
exists  magnetic  force  of  a  definite  amount  acting  in  a  definite 
direction. 


Fig.  68. 

Since  a  magnet  freely  suspended  at  any  point  on  the 
earth's  surface  sets  in  a  definite  direction,  its  north  pole 
always  pointing  approximately  to  the  north  pole  of  the  earth, 
the  magnet  is  in  a  field  of  force  due  to  the  earth ;  the  earth 
itself  exerts  magnetic  force.  Experiments  will  be  described 
later  by  which  the  strength  and  direction  of  this  field  can  be 
determined.  For  the  present  it  is  sufficient  to  bear  in  mind 
its  existence. 

70.  Magnetic  Lines  of  Force.  Suppose  now  that 
at  the  point  P  (Fig.  69)  magnetic  force  is  acting  in  the 


Fig.  69. 

direction  PQ,  that  is  to  say,  a  north  or  positive  pole  placed 
at  P  is  urged  in  the  direction  PQ ;  if  we  move  on  to  $,  a 


69-70]        MAGNETIC   ATTRACTION   AND   REPULSION          113 

point  near  to  P  lying  on  the  line  of  action  of  the  force  at  P, 
the  direction  of  the  force  will  in  general  change ;  let  it 
become  QR.  At  R  a  point  near  Q  the  direction  will  again 
have  altered  to  RS  suppose.  We  thus  have  a  line  PQRS.*. 
made  up  of  a  series  of  short  straight  pieces  PQ,  QR,  RS,  etc. 
which  has  the  property  that 
these  straight  pieces  respec- 
tively each  give  the  direction 
of  the  magnetic  force  at  the 
respective  points  P,  Q,  R.  Now 
if  the  lengths  of  these  pieces 
be  very  much  reduced  we  arrive 
finally  at  a  continuous  curve 
PQRS  (Fig.  70),  and  this  curve 
has  the  property  that  its  direc-  Fig.  70. 

tion  at  each  point  of  its  length 

gives  the  direction  of  the  resultant  magnetic   force   at  that 
point;  it  is  called  a  line  of  magnetic  force. 

We  arrive  thus  at  the  following 


DEFINITION.     A   line   of  Magnetic    Force   is  a  lit 

whose  direction  at  each  point  of  its  length  gives  the  direction  of 
the  magnetic  force   at  that  point. 

Thus  we  may  picture  to  ourselves  the  magnetic  field  as 
permeated  by  the  lines  of  magnetic  force  ;  if  in  any  case  we 
can  map  out  the  lines  of  force  we  can  calculate  the  distribution 
of  the  force  in  the  field. 

The  above  definition  and  explanation  of  a  line  of  magnetic  force 
should  be  compared  with  those  given  in  Section  20  of  lines  of  electrical 
force. 

PROPOSITION  4.  A  small  magnet  placed  in  a  magnetic  field 
sets  itself  with  its  axis  along  the  line  of  force  through  its  centre. 

Suppose  we  place  a  small  magnet  NS  (Fig.  71)  in  the 
field  and  let  it  be  free  to  move  about  an  axis  through  its 
centre  0.  Let  OP  be  the  direction  of  the  magnetic  force 
at  0,  the  direction  that  is  in  which  a  north  pole  at  0  would 
be  urged.  The  north  end  N  of  the  small  magnet  is  pushed  in 
the  direction  OP,  the  south  end  S  pulled  in  the  opposite 
direction;  thus  NS.  is  turned  until  its  axis  points  in  the 

o.  E.  8 


114  MAGNETISM  [CH.  VII 

direction  of  the  force  OP;  the  direction  then  in  which  the 
axis  of  the  small  magnet  rests  is  that  of  the  line  of  force 
through  its  centre. 


Fig.  71. 

It  is  not  necessary  for  the  success  of  this  method  of 
determining  the  direction  of  a  line  of  force  that  NS  should  be 
a  magnet,  if  it  be  an  elongated  piece  of  soft  iron  placed  as  in 
Fig.  71,  then  N  becomes  by  induction  a  north  pole,  S  a  south 
pole  under  the  action  of  the  force,  and  the  results  are  the 
same,  the  direction  in  which  the  soft  iron  sets  is  that  of  the 
line  of  force.  There  is  this  difference  in  the  two  cases  ;  with 
the  magnet  we  can  tell  the  direction  of  the  line  of  force,  it 
will  be  from  the  south  pole  S  to  the  north  pole  N ;  with  the 
piece  of  soft  iron  either  end  may  have  become  a  north  pole  by 
induction,  we  have  nothing  to  tell  us  which  it  is. 

71.  Tracing  Lines  of  Force.  We  may  employ 
either  of  these  methods  to  trace  the  lines  of  force  in  a 
magnetic  field. 

EXPERIMENT  17.  To  trace  the  Lines  of  Force  in  a  magnetic 
field  by  the  use  of  a  small  compass-needle. 

Fasten  a  sheet  of  drawing  paper  on  a  horizontal  drawing 
board,  and  place  a  small  compass  on  the  board,  removing  all 
other  magnets  from  the  neighbourhood.  Make  a  dot  with  a 
pencil  opposite  each  end  of  the  compass-needle.  Move  the 
compass  so  that  its  south  pple  is  over  the  dot  which  was 
opposite  its  north  pole,  and  make  a  dot  opposite  its  north 
pole. 


70-71]        MAGNETIC   ATTRACTION   AND   REPULSION          115 

A  small  compass-needle  pivoted  in  a  brass  cell  between 
two  discs  of  glass  is  convenient  for  this  experiment ;  the  dots 
can  be  seen  through  the  glass.  Continue  this  process  and 
thus  find  a  series  of  positions  indicating  the  direction  of  the 
resultant  force  at  the  successive  points.  Draw  a  line  through 
all  the  dots ;  this  will  be  a  line  of  force.  In  the  same  way 
draw  a  second  line  of  force  on  the  paper.  It  will  be  found 
that  the  two  lines  are  straight  and  parallel,  they  .indicate  the 
direction  in  which  the  magnetic  force  due  to  the  earth  acts 
on  the  magnet,  and  run  north  and  south.  Now  take  a  bar 
magnet  and  place  it  on  the  paper.  Draw  a  line  round  the 


Fig.  72. 

magnet,  and  mark  off  a  number  of  points  on  this  line; 
from  these  points  draw  a  series  -of  lines  of  force,  using  the 
compass  as  before. 

8—2 


116 


MAGNETISM 


[CH.  VII 


Repeat  this  for  different  positions  of  the  magnet.  The 
form  of  the  lines  obtained  which  are  due  to  the  combination 
of  the  force  due  to  the  magnet  and  that  due  to  the  earth 
will  depend  on  the  position  of  the  magnet,  and  on  its  strength. 
If  the  axis  lies  north  and  south,  with  the  north  pole  to  the 
north,  they  will  be  as  in  Fig.  72,  if  the  north  pole  points  to 
the  south  the  lines  of  force  will  be  as  in  Fig.  73.  In  Figure  74 
the  axis  of  the  magnet  is  inclined  to  the  north  and  south  line. 


Fig.  73. 

The  points  P,  P  in  Figures  72  and  74  are  points  at  which 
the  force  due  to  the  magnet  exactly  balances  that  due  to  the 
earth.  They  are  thus  points  of  zero  force. 

The  lines  of  force  due  to  a  combination  of  magnets  can  be 
traced  in  a  similar  manner. 


71] 


MAGNETIC   ATTKACTION    AND   REPULSION 


117 


The  general  forms  of  the  lines  of  force  can  be  more  easily 
observed  by  the  use  of  iron  filings.  The  magnet  or  magnets 
whose  field  is  to  be  explored  are  placed  on  the  table.  Over 


Fig.  74. 

them  is  placed  a  thin  plate  of  glass,  and  on  the  glass  a  sheet 
of  paper.  Iron  filings  ar'e  dusted  on  to  the  paper  through  a 
sieve,  and  the  glass  plate  is  tapped  so  as  to  aid  the  filings  to 
take  up  their  -positions  under  the  magnetic  forces. 

Figures1  75  a  to  e  shew  the  forms  of  the  lines  of  force  due 
to  a  single  magnet,  and  also  to  two  magnets  placed  in  the 
positions  indicated. 


1  These  figures   are  taken   by  the  author's  kind   permission   from 
Watson's  Text-book  of  Physics.     Longmans. 


118 


MAGNETISM 


[CH.  VII 


72.  Tensions  and  Pressures  in  a  Magnetic 
Field.  In  dealing  with  electrostatic  action  we  have  seen  we 
can  explain  the  phenomena  by  supposing  a  tension  to  exist 
along  the  lines  of  force,  combined  with  a  pressure  at  right 
angles  to  them ;  we  can  apply  the  same  idea  to  magnetism. 


Fig.  75  a. 

In  Figures  75  b,  c,  and  e  lines  of  force  are  seen  joining  the 
north  pole  of  either  magnet  to  the  south  pole  of  the  other ; 


Fig.  75  b. 

along  these  lines  there  is  attraction ;   if  we  follow  the  lines 
from  the  north  pole  of  one  magnet  which  start  in  the  direction 


72] 


MAGNETIC    ATTRACTION    AND   REPULSION 


119 


of  the  north  pole  of  the  other,  they  are  apparently  deflected 
from  their  course  by  the  lines  which  issue  from  the  second 
north  pole,  and  the  two  series  of  lines  run  off  together  in 


more  or  less  parallel  directions  :  between  these  two  lines  there 
is  repulsion ;  the  two  like  poles  repel. 


Fig.  75  d. 

We  may  also  examine  by  this  method  the  consequences 
of  placing  a  magnet  in  a   uniform  field.     This  is  shewn  in 


120 


MAGNETISM 


[CH.  VII 


Fig.    72,  in  which   near   the   boundaries   of   the   figure   the 
magnetic  lines  run  uniformly  from  left  to  right;   the  notion 


Fig.  75  e. 

of  tension  along  the  lines  will  explain  how  it  is  that  the 
magnet  NS  tends  to  set  itself  with  its  north  pole  to  the  right 
of  the  figure. 


Figs.  76  a,  76  b. 


72] 


MAGNETIC    ATTRACTION    AND   REPULSION 


121 


If  again  a  sphere  of  soft  iron  be  placed  in  the  field,  the 
lines  of  force  in  its  neighbourhood  are  no  longer  uniformly 
distributed.  They  crowd  into  the  iron  at  one  side  and  leave 
it  again  at  the  other  as  shewn  in  Fig.  76 a ;  the  iron  is  said  to 
have  a  greater  permeability  to  the  magnetic  induction  than 
the  air  which  it  displaces. 

If  the  sphere  be  diamagnetic  the  distribution  of  the  lines 
is  as  in  Fig.  766. 

If  a  ring  of  soft  iron  be  placed  in  the  field  of  a  bar  magnet 
the  effect  is  as  shewn  in  Fig.  77.  There  are  practically  no 
lines  of  force  within  the  ring.  This  explains  how  it  is  that  a 


Fig.  77. 

thick  shell  of  iron  acts  as  a  magnetic  shield,  the  lines  of  force 
from  outside  do  not  enter  the  air  space  in  the  interior  of  the 
shell ;  the  magnetic  force  within  is  much  less  than  it  would 
be  if  the  ring  were  removed. 


CHAPTER   VIII. 

LAWS  OF  MAGNETIC  FOECE. 

73.  Charges  of  Magnetism.     We  shall  find  it  con- 
venient to  speak  of  a  magnet  as  being  charged  with  magnetism. 
In    the  case    of   a  long  thin  magnet  the    charges — the  one 
positive,   the   other   negative — reside   near   the    ends   of   the 
magnet,  and  may  be  considered  as  concentrated  at  two  points, 
the   north  and  south  poles  of  the  magnet  respectively  ;    the 
forces  of  attraction  or  repulsion  observed  between  two  magnets 
arise  on  this  view  from  the  repulsions  or  attractions  between 
the  charges  of  magnetism  in  the  magnets.     A  magnetic  pole 
is  a  point  at  which  a  charge  of  magnetism  is  concentrated, 
and  the  force  between  two  poles  is  the  force  between  the  two 
charges  concentrated  at  these  poles.     This  charge  is  known  as 
the  strength  of  the  pole. 

We  may  also  view  a  magnetic  pole  as  a  centre  from  which 
lines  of  force  diverge  or  to  which  they  converge,  and  picture 
to  ourselves  the  action  between  the  poles  as  due  to  the 
tensions  and  pressures  set  up  in  their  neighbourhood  by  the 
presence  of  these  lines  of  force.  According  to  this  view  the 
strength  of  a  pole  will  be  proportional  to  the  number  of  lines 
of  force  which  diverge  from  it.  This  latter  method  will 
probably  lead  us  further  in  any  attempt  to  explain  the  cause 
of  magnetic  force,  the  former  lends  itself  more  readily  to 
calculation. 

74.  Law   of  Magnetic   Force.     Coulomb  was  the 
first  to  determine  the   law  of   force  between  two  magnetic 


73-75]  LAWS   OF   MAGNETIC   FORCE  123 

poles.  He  shewed  by  means  of  the  torsion  balance  that  the 
force  between  two  poles  of  strengths  m,  m  placed  at  a  distance 
r  apart  is  proportional  to  mm'/r2. 

When  the  torsion  balance  is  used  for  magnetic  experi- 
ments the  light  shellac  needle  which  carries  the  pith  balls 
(shewn  in  Fig.  51)  is  replaced  by  a  long  thin  magnet,  and  the 
charged  conductor  by  a  second  such  magnet.  The  instrument 
is  then  used  in  the  same  way  as  in  the  electrical  investiga- 
tion except  that  a  correction  is  required  for  the  effect  due  to 
the  earth's  field.  The  law  of  force  however  is  more  accurately 
verified  by  an  experiment  due  to  Gauss  described  in  Section  94; 
the  interest  of  the  torsion  balance  is  chiefly  historical. 

Thus  it  follows  from  these  experiments  that  the  force 
between  two  magnetic  poles  m,  m  follows  the  same  law  as 
that  between  two  electrical  charges  e,  e.  If  we  measure 
our  various  quantities  in  proper  units  we  may  write 

mm'  , 

F  —  —  r—  dynes. 
ir 

75.  Unit  Magnetic  Pole.  Suppose  now  we  have 
two  exactly  equal  poles,  so  that  m  is  equal  to  m,  and  that 
they  are  placed  at  a  distance  of  1  cm.  apart,  so  that  r  is  equal 
to  1  ;  suppose  further  it  is  found  that  the  force  of  repulsion 
is  1  dyne,  so  that  F  is  I.  Then  substituting  in  the  above 
equation  we  have 


or  in  —  ±  1. 

Hence  each  pole  must  be  of  unit  strength  and  we  obtain  the 
following  _ 

DEFINITION  OF  UNIT  POLE.  A  Magnetic  Pole  has  Unit 
Strength  when  it  repels  an  equal  pole  placed  at  a  distance  of 
1  centimetre  with  a  force  of  1  dyne.  *+~~~^-*  crx_-*l^_. 

Thus  with  this  definition  of  unit  strength  we  may  say 
that  there  is  a  force  of  repulsion  between  two  poles  of  strengths 
m,  m  placed  at  a  distance  of  r  centimetres  apart  of  mm/r* 
dynes. 


124  MAGNETISM  [CH.  VIII 

If  one  of  the  two  m,  m'  is  negative,  this  force  of  repulsion 
is  negative,  that  is,  the  force  is  one  of  attraction. 

In  strictness  just  as  in  the  corresponding  equation  in  electricity  a 
quantity  K  is  introduced  which  represents  the  action  of  the  dielectric 
medium,  so  here  we  ought  to  introduce  a  quantity — /A — to  represent  the 
magnetic  action  of  the  space  between  the  two  poles  ;  since  however  the 
effects  of  all  media  except  iron  and  to  a  less  degree  nickel  and  cobalt  are 
very  nearly  indeed  the  same  as  that  of  air,  /A,  the  permeability  of  the 
medium,  as  it  is  called,  is  assumed  to  be  unity  and  not  explicitly  intro- 
duced into  the  equation  which  ought  more  accurately  to  be  written 

F_  l»»m' 

fJL      f2      ' 

The  fact  that  /A  is  unity  for  most  materials  is  proved  by  the  observation 
that  the  force  between  two  magnetic  poles  is  not  altered  by  placing 
various  materials  between  the  two.  Compare  Section  27. 

76.  Total  Magnetic  Charge  of  a  Magnet.     We 

have  said  that  the  magnetic  action  of  a  magnet  may  in  many 
cases  be  represented  as  due  to  two  opposite  poles,  the  one 
placed  near  its  north-pointing  end,  the  other  near  its  south- 
pointing  end ;  such  a  magnet  is  called  a  solenoidal  magnet. 

It  follows  accurately  as  the  result  of  experiments  described 
in  Section  90,  that  the  strengths  of  these  two  poles  are  equal ; 
if  the  north  pole  contains  m  units  of  magnetism  the  south 
pole  contains  —  m.  Thus  the.  total  quantity  of  magnetism  in 
the  magnet  is  zero. 

77.  Solenoidal  Magnet.    In  dealing  with  a  solenoidal 
magnet  the  following  definitions  of  the  terms  magnetic  axis 
and  magnetic  moment  will  be  useful. 

DEFINITION.  The  line  joining  the  poles  of  a  magnet  is 
called  its  Magnetic  Axis. 

/"^DEFINITION.  The  product  of  the  strength  of  either  pole  of 
a  magnet  into  the  distance  between  the  poles  is  called  the 
Magnetic  Moment  of  the  Magnet. 

DEFINITION.  The  ratio  of  the  magnetic  moment  of  a 
magnet  to  its  volume  is  known  as  the  Intensity  of  Magneti- 
sation of  the  magnet. 

More  general  definitions  of  theTirst  two  of  these  quantities 
applicable  to  all  magnets  will  be  found  in  Section  90. 


75-79]  LAWS   OF  MAGNETIC   FORCE  125 

78.  Resultant    Magnetic   Force.     Intensity   of 
the  Field.     If  a  unit  positive  pole  be  placed  at  any  point 
of  a  magnetic  field"  it  will  experience  a  certain  force.     This 
force  is  known  as  the  magnetic  intensity  of  the  field  or  the 
resultant  magnetic  force  at  that  point  of  the  field.     Let  it  be 
R  dynes.     Then  a  pole  of  strength  m   placed  at  that  point 
will  be  acted  on  by  a  force  of  Rm'  dynes. 

If  for  example  the  field  is  due  to  a  single  pole  of  strength 
m  at  a  distance  of  r  centimetres,  since  the  force  between  the 
two  poles  m,  m'  is  mm'/r2,  it  is  clear  that  R  is  equal  to  m/r2, 
or  the  intensity  of  the  field  at  a  distance  r  from  a  pole  m  is 

m/r2. 

We  may  compare  these  results  with  those  at  which  we 
have  already  arrived  in  the  case  of  a  charge  of  electricity  in 
Section  29. 

79.  Magnetic  Potential.     In  dealing  with  electro- 
statics we   have  explained  what   is  meant   by  the  electrical 
potential  and  how  it  is  measured ;  similarly  in  magnetism  we 
have  to  consider  the  magnetic  potential  or  magnetic  pressure 
at  a  point,  this  is  measured  by  the  work  done  in  bringing  a 
unit  magnetic  pole  up  to  the  point. 

We  thus  have  the  following  definitions  of  Resultant  Mag- 
netic Force  and  of  Magnetic  Potential. 

^DB^JNITI^F^  The   Resultant    Magnetic    Force   at  a 
point  is  the  force  on  a  unit  magnetic  pole  placed  at  the  point. 

DEFINITION.  The  Magnetic  Potential  at  a  point  is  the 
work -done' in  bringing  a  unit  magnetic  pole  from  beyond  the 
boundaries  of  the  field  up  to  the  point. 

These  definitions  should  be  compared  with  the  corresponding 
ones  inJSlectrostatics  (Section  31). 


CHAPTER   IX. 

EXPERIMENTS  WITH   MAGNETS. 

80.  Methods  of  Magnetisation.  We  have  seen  that 
a  piece  of  iron  or  steel  can  be  magnetised  by  contact  with 
another  magnet,  and  that  in  the  case  of  a  steel  bar  the 
magnetism  so  produced  is  in  great  measure  permanent.  Power- 
ful magnets  are  now  generally  produced  by  the  action  of  an 
electric  current  in  a  manner  which  will  be  described  later 
(Electromagnetism).  The  following  experiments  illustrate 
some  of  the  older  methods  of  making  a  magnet  from  a  piece  of 
steel. 

EXPERIMENT  18.     To  magnetise  a  piece  of  steel. 

(i)  By  single  touch.  Verify  that  the  piece  of  steel  which 
may  conveniently  be  a  piece  of  a  knitting-needle  is  un- 
magnetised.  Take  a  bar  mag- 
net and  determine  by  the  aid 
of  a  compass-needle  as  in  Ex- 
periment 17,  which  is  its  north 
end. 

Draw  the  north  pole  of  the 
bar  magnet  from  one  end  A  to 
the  other  end  B  of  the  piece  of 
steel,  Figure  78.  Repeat  this 
several  times,  beginning  each 
time  at  the  same  end.  Present  Fig.  78. 

one  end  of  the  piece  of  steel  to 

the  compass-needle  and  observe  that  it  is  magnetised,  and 
that  the  end  A  from  which  the  north  pole  was  drawn  is  its 
north  pole,  while  B  is  its  south  pole. 


80-81]  EXPERIMENTS   WITH   MAGNETS  127 

(ii)  By  divided  touch.  For  this  two  bar  magnets  of  equal 
strength  are  required.  Put  the  piece  of  steel  on  the  table. 
Place  the  north  pole  of  one  bar  magnet  in  contact  with  the 
south  pole  of  the  other,  on  the  centre  of  the  piece  of  steel. 
Draw  the  two  magnets  apart  to  the  ends  A,  B  of  the  (Fig.  79) 


Fig.  79. 

piece  of  steel.  Repeat  this  several  times.  The  piece  of  steel  is 
thus  magnetised  ;  on  presenting  it  to  the  compass-needle  it  will 
be  found  that  the  end  A  to  which  the  north  pole  of  one  of  the 
magnets  was  drawn  has  become  a  south  pole,  while  the  end  B 
to  which  the  south  pole  was  drawn  has  become  a  north  pole. 

The  effect  in  this  case  is  increased  by  allowing  the  ends 
Ay  B  of  the  bar  which  is  to  be  magnetised,  to  rest  on  the 
north  and  south  poles  respectively  of  two  other  equally  strong 
magnets  as  shewn  in  Figure  80. 


Fig.  80. 

81.  Magnetic  Batteries.  In  the  methods  of  mag- 
netisation just  described  it  is  chiefly  the  outer  layers  of  the 
steel  that  are  affected.  More  powerful  magnets  are  sometimes 
constructed  by  taking  a  number  of  plates  of  thin  steel,  mag- 
netising each  separately,  and  then  building  them  up  into  a 
permanent  magnet.  In  some  cases  two  pieces  of  soft  iron  are 
placed  as  shewn  in  Fig.  81,  over  each  end  of  the  bar  ;  these 
become  magnetised  by  induction  and  serve  to  keep  the  ends 
of  the  steel  plates  together. 


128  MAGNETISM  [CH.  IX 

Another  form  of  magnet  is  that  used  in  the  Kew  pattern 
instruments  for  measuring  the  strength  of  the  magnetic  force 
due  to  the  earth;  in  these  instruments  the  magnets  are  hollow 
steel  cylinders. 


Fig.  81. 

82.  Demagnetisation  due  to  the  Ends.     In  the 

case  of  a  compound  magnet  such  as  that  just  described,  the 
magnetic  moment  of  the  whole  is  far  from  being  the  sum  of 
the  moments  of  its  parts ;  for  each  bar  acts  inductively  on  its 
neighbours,  and  developes  in  them  a  magnetisation  opposite  to 
its  own;  by  this  action  the  resultant  magnetism  is  reduced;  a 
similar  action  occurs  in  any  magnet,  for  let  N 'St  Fig.  82,  be  a 
magnet  and  P  any  point  in  it, 
then  at  P  there  will  be  a  mag- 
netic   force    acting   from  N  to-       N 
wards  S  tending  to  produce  by 
induction    a    magnet    with    its  Fig.  82. 

north   pole   towards  S,  and  its 

south    towards    N,  a  distribution   of   magnetism  opposite  to 
that  originally  existing. 

83.  Action  of  Keepers.     In  consequence  of  this  de- 
magnetising action  there  is  a  tendency  for  any  magnet  to  lose 
its  magnetism,  and  this  is  increased  if  the  magnet  be  subject 
to  a  jar  or  shock  of  any  kind. 

To  reduce  the  action,  permanent  magnets  are  often  fitted 
with  a  keeper,  a  bar  of  soft  iron  placed  so  as  to  connect  the 
poles.  Thus  in  the  case  of  a  bar  magnet,  two  magnets  are 
kept  together  with  their  poles  arranged  as  in  Fig.  83 ;  the 
two  poles  N,  S'  are  connected  by  one  piece  of  soft  iron  AB  and 
the  two  JV',  S  by  a  second  piece  CD.  By  this  means  the  de- 
magnetising force  in  the  magnets  is  reduced,  for  the  north  pole 
N  induces  a  south  pole  at  A,  and  the  effect  of  this  at  any  point 
P  of  the  magnet  is  opposite  and  nearly  equal  to  that  of  the 
original  north  pole. 


81-84]  EXPERIMENTS   WITH   MAGNETS  129 

In  the  case  of  a  horse-shoe  magnet  a  single  piece  of  soft 
iron  is  placed  across  the  poles,  and  the  same  effect  is  produced. 

We  have  already  seen  that  when  a  piece  of  soft  iron  is 
placed  in  a  magnetic  field  the  lines  of  force  are  concentrated 
through  the  iron.  When  a  single  bar  magnet  is  alone  lines  of 
force  diverge  from  its  north  pole,  and  pass  round  through  the 
air  to  its  south  pole,  many  however  return  through  the  magnet 
itself;  the  demagnetising  action  is  due  to  these.  When  a 
second  bar  magnet  is  placed  near  as  in  Fig.  83,  the  poles  being 
v 


Fig.  83. 

reversed  in  the  two,  even  before  the  keepers  are  put  on  many 
lines  of  force  pass  from  N  to  S'  across  the  air  gap  and  the 
number  through  the  magnet  from  N  to  S  is  reduced ;  by 
placing  the  keepers  AB  and  CD  in  position  practically  all  the 
lines  are  made  to  flow  through  the  soft  iron  from  N  to  S' ;  the 
number  returning  to  S  through  the  magnet  is  very  small  indeed. 

84.  Susceptibility.  The  magnetisation  produced  in  a 
steel  bar  by  a  given  magnetising  force  depends  very  greatly  on 
the  nature  and  temper  of  the  steel,  and  can  be  largely  modified 
by  heating  and  annealing. 

A  bar  of  soft  iron  can  be  much  more  strongly  magnetised 
by  a  given  force  than  a  bar  of  steel,  the  steel  however  retains 
much  more  of  its  magnetism  than  the  iron  unless  indeed  the 
latter  be  very  specially  treated.  The  ratio  of  the  intensity  of 
magnetisation  produced  by  a  given  force  to  that  force  is  known 
as  the  susceptibility,  thus  soft  iron  has  a  higher  susceptibility 
than  steel ;  the  magnetism  which  remains  after  the  magnetising 
force  has  been  removed  is  spoken  of  as  permanent  or  residual 
in  contrast  to  the  temporary  magnetism  which  disappears  with 
the  force.  Bodies  which  retain  a  large  portion  of  their 
magnetism,  as  permanent  or  residual,  are  said  to  have  a  large 

G.  E.  9 


130  MAGNETISM  [CH.  IX 

"coercive  force."  In  steel,  then,  the  coercive  force  is  high, 
and  depends  greatly  on  the  temper  of  the  steel. 

For  permanent  magnets  the  steel  is  usually  annealed  to  a 
blue  temper,  and  may  with  advantage  contain  a  small  quantity 
of  tungsten  or  molybdenum. 

85.  Influence  of  Temperature.    When  a  steel  mag- 
net  is  heated   to   a   moderate   extent   it    loses   some   of   its 
magnetism.     Part  of   this  it  usually   recovers  on   cooling — 
unless  it  has  been  raised  to  too  high  a  temperature.     If  it  be 
raised  to  a  bright  red  heat  and  allowed  to  cool,  when  free  from 
the  action  of  magnetic  force,  it  loses  all  its  magnetism.     In 
the  case  of  moderate  heating  the  permanent  loss  on  cooling 
depends  on  the  extent  of  the  heating,  and  on  the  previous 
history  of  the  magnet;  if  for  example  a  magnet  be  "aged"  by 
repeatedly  heating  it  up  to  say  100°  C.  magnetising  it  as  com- 
pletely as  possible  when  at  that  temperature,  and  allowing  it 
to  cool  slowly,  then  after  a  long  succession  of  such  changes  its 
permanent  magnetism  is  not  seriously  affected  by  heating  to  a 
lower  temperature  than  that  reached  in  the  "  ageing  "  process. 
For  a  magnet  so  treated  the  magnetic  moment  decreases  uni- 
formly as  the  temperature  rises,  increasing  again  with  a  falling 
temperature  to  its  original  value  ;   the  relation  between  the 
magnetic  moment  and  the  temperature  is   expressed   by  the 
equation    M  =  M0(l—qt)    where    J/0   is   the    moment   at   the 
freezing  point,  M  the  moment  at  a  temperature  of  t°,  and  q 
the    coefficient   of    change   of    moment   for    each   degree    of 
temperature. 

86.  Maximum  Strength  of  the  Pole  of  a  Magnet. 

The  strength  of  the  poles  of  an  ordinary  bar  magnet  having  a 
length  of  say  10  centimetres  or  more,  and  a  cross  section  of 
about  1  square  centimetre,  may  be  from  100  to  200  units;  with 
a  long  thin  bar  the  strength  per  unit  area  of  the  cross  section 
of  the  bar  may  be  much  greater  than  this.  The  magnetic 
moment  of  a  bar  10  centimetres  long  with  a  pole  strength 
of  200  would  be  10  x  200  or  2000. 

Since  the  volume  of  such  a  bar  would  be  10  cubic  centi- 
metres the  intensity  of  its  magnetisation  would  be  200. 

87.  Theory  of  Magnetisation.     We  have  already 
stated  that  if  a  magnet  be  broken  into  two  pieces  each  part 


84-88]  EXPERIMENTS   WITH   MAGNETS  131 

becomes  a  magnet ;  this  is  readily  shewn  by  magnetising  a 
thin  knitting  needle.  On  breaking  it  into  two  and  dipping 
the  ends  of  either  half  into  iron  filings,  the  filings  adhere  to 
both  ends ;  if  each  of  the  two  halves  be  now  broken  into  two 
the  portions  into  which  they  are  divided  have  each  two  poles. 
This  process  can  be  continued  as  long  as  the  pieces  of  steel 
are  large  enough  to  be  broken.  ^  We  are  thus  led  to  infer  that 
magnetism  may  be  a  molecular  phenomenon,  and  that  the 
magnetic  forces  observed  are  the  resultants  of  those  due  to 
the  molecular  magnets  which  make  up  the  whole. 

Thus,  suppose  we  have  a  large  number  of  similar,  equal 
and  equally  magnetised  small  magnets.  Let  them  be  placed 
in  a  row  as  in  Fig.  84,  in  such  a  way  that  the  north  pole  of 
one  is  adjacent  to  the  south  pole  of  the  next. 

Nf 


s|n       s|n        s| n        s [n      s|n     s|n       |^ 

Fig.  84. 

Then  except  at  the  two  ends  the  magnetic  repulsion  of 
any  north  pole  n"  will  be  neutralised  by  the  attraction  of  the 
adjacent  south  pole  s  ;  the  resultant  force  due  to  the  com- 
pound bar  will  consist  of  a  repulsion  from  one  end  N  combined 
with  an  attraction  to  the  other  end  S ',  the  bar  will  behave  as 
a  simple  solenoidal  magnet.  We  may  thus  picture  to  our- 
selves the  effect  of  an  actual  magnet  as  due  to  the  magnetism 
of  its  molecules. 

Again,  consider  a  test-tube  full  of  iron  filings ;  the  tube  is 
magnetic,  but  it  is  not  a  magnet.  Bring  it  however  into  a 
magnetic  field,  so  that  its  lower  end,  for  example,  is  near  the 
north  pole  of  a  strong  magnet,  and  tap  it  gently ;  on  removing 
it  we  shall  find  that  the  lower  end  has  become  a  south  pole. 
In  the  magnetic  field  the  filings  have  set  themselves  so  that 
their  axes  are  along  the  lines  of  force,  just  as  we  saw  them 
do  when  observing  the  lines  of  force  on  the  glass  plate,  and 
the  iron  has  thus  become  a  magnet.  On  shaking  the  tube  up 
again  it  ceases  to  be  a  magnet. 

88.  Molecular  Magnets.  In  considering  this  mole- 
cular magnetism,  the  question  arises  whether  the  molecules 

9—2 


132  MAGNETISM  [CH.  IX 

of  iron  are  or  are  not  permanent  magnets.  If  they  are 
permanent  magnets,  the  process  of  magnetisation  will  consist 
in  arranging  them  so  that  their  axes  all  point  in  the  same 
direction ;  if  they  are  not  permanent  magnets,  the  process 
will  consist  firstly  in  magnetising  each  molecule,  and  then  in 
arranging  the  assemblage.  If  the  first  assumption  be  the 
true  one,  and  each  molecule  of  a  magnetic  substance  is  itself 
a  magnet,  we  have  to  explain  why  every  piece  of  iron  is  not  a 
magnet.  This  explanation  was  given  by  Weber,  who  pointed 
out  that  the  axes  of  the  molecular  magnets  would  set  in  all 
directions,  so  that  on  the  whole  there  would  be  no  magnetic 
force  from  a  piece  of  iron,  while  Professor  Ewing  extended 
this  by  pointing  out  that  under  the  mutual  forces  between 
the  magnets  any  large  assemblage  of  small  magnets  would  set 
themselves  so  as  to  form  closed  circuits,  and  would  thus  have 
no  external  effect.  The  north  pole  of  one  magnet  attracts 
the  south  pole  of  one  of  its  neighbours,  while  its  own  south 
pole  is  attracted  by  some  other  north  pole.  The  lines  of  force 
thus  form  closed  circuits  within  the  magnets ;  few  if  any 
escape  beyond  the  assemblage. 

Professor  Ewing  illustrated  this  by  experimenting  with  a 
large  number  of  small  compass-needles,  resting  on  pivots, 
arranged  in  regular  order  on  a  horizontal  board  ;  the  com- 
pass-needles set  themselves  in  the  various  manners  shewn  in 
Fig.  85  ;  if  they  are  temporarily  disturbed  from  these  positions 


\     /\  ,\\ 

lx  '-rj  \\  \ 


Fig.  85. 

they  fall  into  some  other  having  similar  properties,  and  so 
long  as  they  are  free  from  external  force  they  as  a  rule  exert 
no  magnetic  force  themselves.  He  was  able  to  shew  that 
by  varying  the  grouping  of  these  small  magnets  many  of  the 
magnetic  properties  of  iron  and  steel  can  be  very  closely 
imitated. 


88] 


EXPERIMENTS   WITH    MAGNETS 


133 


Thus  suppose  that  by  means  of  some  impressed  mag- 
netic force  the  old  grouping  is  broken  up  and  a  new  one 
established ;  if  this  new  grouping  be  very  stable  it  will 
continue  when  the  magnetic  force  is  withdrawn :  the  assem- 
blage resembles  a  steel  magnet  \  if  the  stability  of  the  new 
grouping  be  very  small  then  on  removing  the  magnetic  force  the 
magnets  rearrange  themselves  on  the  slightest  disturbance,  so 


Fig.  86 l. 

as  to  'produce  no  external  field,  the  assemblage  is  like  a  piece 
of  soft  iron.  All  these  changes  can  go  on  under  the  mutual 
magnetic  forces  between  the  molecu]es  ;  the  assumption  of  a 
force  arising  from  friction  or  some  similar  action  tending  to 
hold  them  in  their  positions  is  not  needed. 


Fig.  87 !. 

1  These  Figures  are  taken  by  the  author's  kind  permission  from 
Watson's  Text-book  of  Physics.     Longmans. 


134  MAGNETISM  [CH.  IX 

We  may  also  examine  the  action  of  a  number  of  small 
magnets  by  tracing  by  means  of  iron  filings  the  lines  of  force 
they  produce ;  if  the  magnets  be  arranged  in  order  with  all 
their  like  poles  pointing  the  same  way  as  in  Fig.  86,  there  is 
a  very  distinct  external  field  observable ;  if  however  they  be 
allowed  to  take  up  the  kind  of  positions  shewn  in  Fig.  87, 
v  the  external  field  is  very  small ;  we  suppose  then  that  in  an 
unmagnetised  piece  of  iron  the  molecules  are  arranged  as  in 
Fig.  87.  When  the  iron  is  magnetised  they  are  as  in  Fig.  86. 

If  the  arrangement  is  stable  so  that  it  is  retained  after  the 
magnetising  force  is  removed,  we  have  the  case  of  a  permanent 
magnet,  if  however  it  is  unstable  the  iron  is  only  temporarily 
magnetised. 

89.    Magnetic  Force  due  to  a  Closed  Cycle.    The 

fact  that  a  closed  ring  of  magnetic  particles  exerts  no  external 
force  is  easily  shewn  by  experiments  with  a  piece  of  watch- 
spring.  On  magnetising  it  and  dipping  it  into  iron  filings 
they  adhere  most  strongly  near  the  ends ;  along  the  length  of 
the  spring  very  few  are  visible.  Now  bend  the  spring  so  that 
its  north  polar  end  is  brought  into  contact  with  the  south 
polar  end,  and  holding  it  in  this  position  again  cover  it  with 
iron  filings.  Very  few  adhere  to  it,  the  ring  behaves  almost 
as  though  it  were  unmagnetised. 


CHAPTER  X. 


MAGNETIC   CALCULATIONS. 


9O.  Magnet  in  a  Uniform  Field.  In  a  number  of 
cases  we  can  calculate  the  force  between  two  magnets,  or  the 
force  on  one  magnet  due  to  a  given  field ;  the  simplest  is  that 
in  which  the  field  is  uniform. 

Let  us   suppose   that  the   strength   of   the   uniform   field 
is  H  and  that  it  acts  in  the  direction  AB,   Fig.   88.     H  is 
then   the    force    with    which    a 
pole  of  unit  strength  would  be 
urged  in  the  direction  from  A 
to  B.     The  force  on  a  pole  m 
would  be  mH,   that  on   a  pole 

—  m   would    be    —  mH,    or   m/f 
acting  from  B  to  A. 

A  magnet  has  two  poles,  let 
us  suppose  m  is  the  strength  of 
one  pole,  -  m  that  of  the  other. 
Then  the  force  on  the  one  pole 
will  be  mff,  that  on  the  other 

—  m'//,    and    the   force   on    the 
magnet  will  be  the  resultant  of 
these. 

We  shall  see  directly  that  in 
is  equal  to  m'.  Fig<  88> 

Now    since    we    have     two 

parallel  forces  acting  on  the  magnet,  the  resultant  consists 
of   a  force  mlf  —  m'H,   or  (m-m')H  in  the  direction  of  77, 


136  MAGNETISM  [CH.  X 

and  a  couple  which  tends  to  turn  the  magnet  until  its  axis 
points  in  the  direction  of  H. 

We  can  shew  however  by  experiment  that  in  a  case  such 
as  this  the  resultant  is  a  couple  only,  the  resultant  force  is 
zero;  from  this  it  follows  that  m  is  equal  to  ra',  or  the 
strengths  of  the  two  poles  of  a  magnet  are  equal. 

The  experiment  is  most  easily  performed  when  the  earth's 
field  is  taken  as  the  uniform  field. 

EXPERIMENT  1 9.  To  shew  that  the  earth's  magnetic  field  is 
directive  only,  and  that  therefore  the  quantities  of  positive  and 
negative  magnetism  in  a  magnet  are  equal. 

Place  a  bar  magnet  on  a  circular  block  of  wood  and  allow 
it  to  float  in  a  large  vessel  of  water;  bring  the  floating 
magnet  to  rest  in  some  position  in  which  its  axis  does  not 
point  north  and  south,  and  then  gently  remove  the  hands 
without  disturbing  the  magnet.  When  set  free  it  will  be 
found  that  the  block  on  which  the  magnet  rests  turns  round, 
but  that  it  does  not  acquire  a  movement  of  translation  through 
the  water ;  it  follows  from  this  that  there  is  a  couple  on  the 
magnet,  but  no  resultant  force. 

Now  if  we  call  m  the  whole  quantity  of  positive,  m  the 
whole  quantity  of  negative  magnetism  in  the  magnet,  the 
resultant  force  in  a  field  H  is  mH—m'ff,  and  since  this  is 
zero  we  must  have  m  =  m.  That  is  the  amounts  of  positive 
and  negative  magnetism  in  any  magnet  are  equal. 

PROPOSITION  5.  To  find  the  couple  on  a  magnet  in  a 
uniform  field. 

Consider  first  a  simple  magnet,  Fig.  89,  having  a  north 
pole  N,  at.  which  m  units  of  magnetism  are  concentrated,  and 
a  south  pole  S  with  -  m  units. 

Let  0  be  the  centre  of  the  magnet,  2Z  its  length,  and  let 
the  axis  SN  make  an  angle  0  with  AOB,  the  direction  of  the 
force  H. 

Through  JV  and  S  draw  LN  and  SK  parallel  to  the  direc- 
tion of  //,  and  through  0  draw  KOL  perpendicular  to  AOB. 

Then  the  forces  we  have  to  consider  are  mil  through  N, 
acting  along  LN,  and  mH  through  S  along  KS. 


90] 


MAGNETIC    CALCULATIONS 


137 


Taking  moments  about  0  we  have  for  the  moment  of  the 
resultant  couple  mH  x  OL  +  mH  x  OK. 


mH 


Now  OL  =  ON  sin6  =  l  sin  0, 

and  OK  =  OS  sin  6  =  I  sin  0. 

Hence  the  moment  required 

=  2m£H  sin  0  =  MH  sin  0, 
if  M  is  the  moment  of  the  simple  magnet. 

Now  we  may  look  upon  any  magnet  as  compounded  of  a 
series  of  such  simple  magnets.  The  resultant  action  on  each 
of  these  will  be  a  couple,  hence  the  whole  resultant  is  a  couple 
which  we  may  write  as  before  MHsinO,  where  M  is  the 
moment  of  the  magnet,  and  0  the  angle  which  its  axis  makes 
with  the  direction  of  H. 

It  is  clear  that  the  couple  is  greatest  when  0  is  90°,  for 
then  sin  0  =  1.  In  this  case  the  axis  of  the  magnet  is  at  right 
angles  to  the  direction  of  H. 

The  couple  is  zero  when  the  axis  coincides  with  the  direc- 
tion of  H,  for  then  sin  0  is  zero. 

These  results  are  obvious. 

Again,  if  //  is  unity  and  0  is  90°,  the  value  of  the  couple 
\aM. 


138 


MAGNETISM 


[CH.  X 


We  thus  arrive  at  the  result  that  the  magnetic  moment  of 
a  magnet  is  measured  by  the  maximum  couple  it  can  experience 
when  placed  in  a  field  of  unit  strength,  and  we  thus  obtain 
two  somewhat  more  complete  definitions  of  the  terms  magnetic 
axis  and  magnetic  moment  than  those  already  given. 

DEFINITION.  Suspend  a  magnet  by  its  centre  of  gravity  in 
a  uniform  magnetic  field.  Then  it  will  be  found  that  there  is 
a  line  through  the  centre  of  gravity  which  always  sets  in  a  fixed 
direction  in  space.  This  line  is  called  the  Magnetic  Axis  of 
the  magnet. 

DEFINITION.  It  is  found  that  a  magnet  placed  in  a  uni- 
form field  is  in  general  acted  on  by  a  couple.  The  ratio  of  the 
'moment  of  this  couple  when  it  is  a  'maximum  to  the  strength  of 
the  field  is  called  the  Magnetic  Moment  of  the  Magnet. 

91.     Magnetic  Force  due  to  a  Simple  Magnet. 

Let  NOS,  Fig.  90,  be  a  simple  magnet  and  P  any  point  near 

at  which  the  magnetic  force  is 

required.  U 

Let     OP  =  r 


Let  m  and  —  m  be  the  pole 
strengths  and  2£  the  length  of 
the  magnet.  Then  the  force  at 
P  is  the  resultant  of  the  forces 
due  to  m  at  N  and  —  m  at  S. 

The  force  due  to  N-  is 
mjPN*  acting  along  NP,  that 
due  to  S  is  m/SP*  acting  along 
PS. 

Produce  NP  and  in  it  take 
PA  equal  to  m/PN*;  in  PS 
take  P.5  equal  to  mjSP*  and  complete  the  parallelogram 
APBC.  Then  the  diagonal  PC  represents  the  resultant 
force. 

We  can  find  a  mathematical  expression  for  this  resultant  in  terms  of 
r,  6  and  the  known  quantities  m  and  I,  but  this  becomes  complicated 
except  in  some  special  cases,  e.g.,  when  P  is  on  the  axis  of  the  magnet 
so  that  0  is  zero  or  when  OP  is  at  right  angles  to  the  axis  so  that  PON 
is  90°. 


Fig.  90. 


90-91]  MAGNETIC   CALCULATIONS  139 

PROPOSITION  6.     To  find  the  force  due  to  a  simple  magnet 
at  a  point  on  its  axis  produced. 

Let  NS,  Fig.  91,  be  the  magnet,  0  its  centre  and  P  the 
point  on  ON  produced  at  which  the  force  is  to  be  found. 


Let 

Then  the  two  forces  due  to  N  and  S  act  in  the  same 
straight  line  but  in  opposite  directions  and  the  resultant  R 
is  their  difference. 

m          m 
Hence  R  = 


NP*     SP* 

m  m 


(r-lf      (r 


4mrl  2  Mr 

=  p~l  py  =  (r-  -I2)2 ' 

We  may  put  the  expression  into  the  form 

2M 


In  many  cases  I  is  so  small  compared  with  r  that  (l/r)*  may  certainly 
be  neglected  while  it  may  also  happen  that  to  the  accuracy  with  which 
we  are  working  2i2  may  be  neglected  compared  with  r2,  e.g.  if  1  =  2  cm. 
and  F  =  20cm.  then  2Ja/r3=  '02,  and  if  we  do  not  mind  an  error  of  2  per" 
cent,  we  may  treat  P  as  zero\compared  with  r2. 

If  we  suppose  I  to  be  so  small  that  we  may  put  (l/r)°  zero 
in  the  expression  for  the  force,  we  have 


140 


MAGNETISM 


[CH.  X 


Thus  we  see  that  in  this  case  the  force  at  a  given  point 
due  to  a  magnet  is  inversely  proportional  to  the  cube  of  the 
distance  of  the  point  from  the  centre  of  the  magnet. 

This  position  of  the  magnet  is  sometimes  known  as  the 
• "  end-on "    position  ;    the   axis   of 
the  magnet  is  directed  end-on  to 
the  point  at  which   the  force  is 
being  calculated.  „    j 

PROPOSITION  7.  To  find  the  force 
due  to  a  simple  magnet  at  a  point 
such  that  the  line  joining  it  to  the 
centre  of  the  magnet  is  perpen- 
dicular to  its  axis. 

Let  NOS  (Fig.  92)  be  the 
magnet,  P  the  point  at  which 
the  force  is  required,  OP  being 
perpendicular  to  NS. 

Then  since  NP  —  PS  the  two  forces  due  to  m  and  -  m  are 
equal,  each  being  m/NP2.  The  resultant  force  bisects  the 
angle  between  NP  produced  and  PS,  it  is  parallel  therefore  to 
NS. 

Resolving  the  forces  parallel  to  NS  we  have  then  for  the 
resultant 


also 


Fig.  92. 


Thus 

But 

Hence 


cos  PNO  «  cos  PSO  =  ON  IN  P. 
ZmOff 


2ml             M 
R  — „  — 


and  if  we  again  take  a  case  in  which  P  is  negligible  compared 
with  r2  we  have 

M 


91-92]  MAGNETIC    CALCULATIONS  141 

Thus  in  this  position  also — which  is  known  as  the  broad- 
side-on  position — the  force  on  a  magnetic  pole  is  inversely 
proportional  to  the  cube  of  its  distance  from  the  centre  of 
the  magnet. 

But  since  we  have 

2M 
R  -  —  3    end-on  position, 

and  R  —  —^  broadside-on  position, 

we  see  that,  at  equal  distances  from  the  centre,  the  force  in 
the  end-on  position  is  twice  as  great  as  in  the  broadside-on 
position. 

The  expressions  we  have  just  found  represent  the  magnetic 
intensities  or  forces  on  a  unit  pole  at  the  two  points ;  if  a  pole 
of  ^strength  m  be  placed  at  either  point,  the  forces  acting  on 
this  pole  will  be  respectively 

2Mm'/r>   and    Jfm'/r3. 

92.    Forces   on   one  Magnet  due  to   a   Second. 

To  find  the  force  on  a  second  simple  magnet  we  must  re- 
member that  it  has  two  poles  ;  it  will  be  necessary  to  calculate 
the  force  on  each  pole  and  to  find  their  resultant,  and  the 
process  is  complex.  When  however  the  second  magnet  is 


fMi^^MPMMfc- 


Fig.  93. 

small  and  at  some  distance  from  the  first  we  may  simplify  the 
calculation    greatly;    for    though   the   field   due  to   the  first 


142  MAGNETISM  [CH.  X 

magnet  is  not  strictly  uniform,  if  N'9  S'  (Fig.  93),  the  poles 
of  the  second  magnet,  are  near  together  and  at  some  distance 
from  N  and  S,  we  may  as  a  first  approximation  treat  the 
force  at  N'  as  equal  to  that  at  S',  each  being  equal  to  that 
at  P  the  centre  of  the  second  magnet. 

Hence  we  have  to  deal  with  the  case  of  a  small  magnet  in 
a  uniform  field,  and  if  R  be  the  strength  of  this  field,  6'  the 
angle  between  the  direction  of  R  and  the  axis  of  the  second 
magnet,  and  M'  the  moment  of  this  magnet,  the  resultant 
action  on  N'S'  is  a  couple  M'fism  6'  tending  to  decrease  6'. 

In  the  end-on  position  the  direction  of  R  is  along  OP, 
0'  is  the  angle  between  PN'  and  OP  produced  and 

2MM'  sin  & 
moment  of  couple  =       — § . 

In  the  broadside-on  position  R  is  at  right  angles  to  OP, 
0'  is  the  angle  between  PN'  and  a  line  through  P  parallel  to 
ffS, 

,          .       MM' sin  6' 
moment  of  couple  =  —  — ,—     • 

These  two  expressions  are  so  important  that  we  will  find 
them  in  another  way. 

PROPOSITION  8.  To  find  the  moment  of  the  couple  acting  on 
a  small  magnet  placed  with  its  centre  at  a  point  on  the  axis  of 
a  second  small  magnet,  assuming  the  distance  between  the  centres 
to  be  large  compared  with  the  dimensions  of  either. 


Fig.  94. 

Let  N'PS'  (Fig.  94)  be  the  second  magnet  placed  with  its 
centre  at  a  point  P  in  the  direction  of  the  axis  SON  of  the 
first  magnet. 

Let  OP  =  r  and  OPS'  =  0',  let  mf  be  the  pole  strength  and 
21'  the  length  of  tjje  second  magnet  and  M'  its  magnetic 
moment. 


92]  MAGNETIC   CALCULATIONS  143 

Then  if  N'S'  is  small  compared  with  OP  the  forces  due 
to  NS  at  N',  8'  and  P  may  be  taken  as  equal,  and  each  is 
equal  to  2M/r3. 

Thus  we  have  to  find  the  resultant  of  two  equal  and 
opposite  parallel  forces  2m'J//r3  at  N'  and  S'  respectively. 

Draw  S'K  parallel  to  NP  and  N'K  to  meet  S'K  in  K  at 
right  angles  to  S'K. 

The  resultant  of  the  two  parallel  forces  is  a  couple  whose 
moment  is 


Now  N'K=N'S'  sin  N'S'K=  21'  sin  &'. 

Thus  moment  of  couple  required 


SJfif'sin? 


PROPOSITION  9.     To  Jind  the  moment  of  the  couple  acting 
on  a  small  magnet  placed  with   its 
centre   so    that    the    line  joining  it  N1 

to    the   centre    of  a    second   magnet     ~~* — 
is  at  right  angles  to  the  axis  of  that 
magnet,    the    distance    between    the 
centres  being   large   compared  with 
the  dimensions  of  either  magnet. 

Here  again  the  forces  at  P,  N' 
and  S'  (Fig.  95)  may  be  treated 
as  equal,  and  equal  to  M/r3. 

This  force  acts  at  right  angles       fmf 

to  PO  on  a  pole  of  strength  m.  S  o  N 

Let  N'K  be  perpendicular  and  Fig.  95. 

S'K  parallel  to  PO. 

Then  the  resultant  required  is  a  couple  whose  moment  is 
m' .  —3  x  S'K. 


144 


MAGNETISM 


[CH.  X 


Substituting  for  S ' K  its  value  21'  sin  &'  we  have 

jrjrriB? 

moment  ot  resultant  couple   =       3 . 

PROPOSITION  10.  A  magnet  hanging  freely  is  deflected  from 
its  equilibrium  position  in  which  its  axis  is  north  and  south  by 
a  distant  magnet.  To  find  the  position  it  will  take  up. 

Let  the  magnet  S'N'  (Fig.  96)  come  to  rest  with  its  axis 
making  an  angle  <£  with  the  north  and  south  line,  and  let  H 
be  the  horizontal  component  of  the  force  due  to  the  earth 
which  acts  on  it. 


Fig.  96. 
The    couple  acting  on  the  magnet  due  to  the    earth  is 


The  couple  due  to  the  distant  magnet  can  be  calculated  if 
we  know  its  position,  let  it  be  M'G.  Then  for  equilibrium 
we  must  have 


or 


sm  <   = 


couple   due  to  distant  magnet 


If  for  example  the  distant  magnet  be  placed  in  the  end-on 
position  and  if  6'  be  the  angle  between  its  axis  and  that  of 
the  deflected  magnet  we  have 

,„„     2MM'  sin  ff 
M  G  =  „ 


Hence 


. 
H  sin  <   = 


3 


1  It  should  be  noted  that  these  expressions  do  not  include  the  effect 
of  the  earth  on  the  magnet. 


92] 


MAGNETIC   CA 


145 


Two  cases  of  interest  occur  in  practice  : 

(i)  When  the  axis  of  the  distant  magnet  points  east  and 
west  (Fig.  97).  In  this  case  PO  is  at  right  angles  to  the 
direction  of  //,  and  &  +  <J>  is  a  right  angle,  and 


sin 


=  cos 


m'H 


Fig.  97. 


Hence 
Whence 


//  sin  <£  =          ., 

2M 

tan  d>  =  , ,  ,  . 


(ii)     When  OP  is  at  right  angles  to  the  deflected  position 
of    the    suspended 
magnet  as  in  Fig.    •51< 
98.     In    this    case 
0'=  90°  and  sin 0  =  1. 


Thus 


2J/ 


These  two  posi- 
tions are  spoken  of 
as  the  tangent  and 
sine  positions  re- 
spectively. 

If  the  distant  mag- 
net be  used  in  the 

broadside-on    position  Fig.   98. 

similar  results  can  be 

obtained    but    the   couple    due    to    it   will    be    MM' sin  0'!^   instead   of 
2MM'  sin  0'/r3,  and  hence  we  shall  obtain 

M 


G.   E. 


10 


146  MAGNETISM  [CH.  X 

and  sin  9!^  =  — -^ 

for  the  two  cases,  0X  being  the  deflexion. 

If  we  call  H  the  controlling  and  R  the  deflecting  field  we 
notice  that  the  tangent  law  holds  when  the  deflecting  field  is 
at  right  angles  to  the  controlling,  while  the  sine  law  holds 
when  the  deflecting  field  is  perpendicular  to  the  magnet. 

We  may  if  we  please  obtain  the  above  formulae  directly 
thus. 

PROPOSITION  1 1.  The  axis  of  a  small  magnet  points  east  and 
west.  A  second  small  magnet  pivoted  at  its  centre  is  placed 
with  its  centre  at  a  point  on  the,  axis  produced.  To  find  the 
position  it  takes  up. 

Let  SON  (Fig.  99)  be  the  first  magnet,  P  the  point  on  the 
axis  produced  at  which  the  centre  of  the  second  magnet 
S'PN'  is. 


nvH 


b± 


Fig.  99. 

If  it  were  not  for  the  presence  of  NS  the  magnet  JVS' 
would  point  north  and  south.  It  is  deflected  from  this 
position  by  the  force  due  to  WS,  let  <j>  be  the  angle  between 
PN'  and  the  undisturbed  position  PK  of  the  second  magnet. 

Draw  N'K  perpendicular  to  OP. 

Let  m'  be  the  strength  of  each  pole  of  A"»S",  M'  its  moment, 
and  let  OP  =  r. 

Let  H  be  the  strength  of  the  field  at  N'  before  the  magnet 
NS  is  brought  near,  and  let  R  be  the  strength  of  the  field 
due  to  NS.  If  OP  is  large  compared  with  N'S'  the  values 
of  R  at  JV't  A$"  and  P  are  equal.  Hence  since  OP  =  r, 


92] f 


MAGNETIC   CALCULATIONS 


147 


Now  N'  is  acted  on  by  two  magnetic  forces  m'H  parallel 
to  KN'  and  m'R  parallel  to  PK  while  S'  is  acted  on  by  equal 
forces  in  opposite  directions.  Since  the  magnet  which  can 
turn  about  P  is  in  equilibrium  the  resultant  of  each  pair  of 
forces  must  pass  through  P,  and  the  forces  are  proportional  to 
the  sides  of  the  triangle  N'KP  to  which  they  are  parallel. 

m'R        KP 
Hence 


or 


m'H      N'K 

tan  </>  =  —  = 


Hr*' 

Thus  the  tangent  law  is  proved. 

PROPOSITION  12.  A  small  magnet  pivoted  at  its  centre  is  de- 
flected by  a  second  magnet.  The  axes  of  the  tivo  magnets  are 
always  at  right  angles,  and  that  of  the  deflecting  magnet  passes 
through  the  centre  of  the  first  magnet.  To  find  the  position  it 
takes  up. 

Let  SON  (Fig.  100)  be  the  first  magnet,  N'PS'  the  second. 
Then  P  -lies  on  ON  produced,  and  N'S1  is  at  right  angles 
to  OP. 


m'H 


Fig.  100. 

Let  R  be  the  strength  of  the  field  due  to  NS  at  P,  since 
N'S'  is  small,  R  is  the  strength  at  N'  arid  /S",  and  its  direction 
is  parallel  to  NP. 

Let  H  be  the  strength  of  the  undisturbed  field  at  P,  and 
let  KN'  be  its  direction,  and  let  the  angle  PN'K  be  <f>. 

10—2 


148 


MAGNETISM 


[CH.  X 


Then  the  forces  on  N'  are  m'H  and  m'R,  while  S'  is  acted 
on  by  equal  forces  in  opposite  directions,  and  for  equilibrium 
we  must  have  the  forces  proportional  to  the  sides  of  the  triangle 
N' KP,  to  which  they  are  parallel. 


Thus 
Also 


m'R  _KP_ 
N'K 


sn 


Hence 


The  formulas  for  the  broadside-on  position  are  proved  similarly ;  the 
figures  will  differ,  and  the  value  of  R  is  M/r3. 

A  more  general  case  is  proved  in  the  same  way. 

Let  a  magnet  N'S '  ( Fig.  101) 
hanging  in  a  field  of  strength  H 
be  deflected  through  an  angle  0 
from  its  equilibrium  position 
by  a  field  R,  and  let  the  direc- 
tion of  R  make  an  angle  ^ 
with  the  original  field. 

Thus  in  the  figure 
PN'K=<f>,     N'KP  =  TT-^. 

Hence  N'PK=\I/-  <j>. 

Therefore 

m'R  _  PK  _     J3inj>_ 
N'K  ~  sin  U  -  0) ' 


or 


Fig.  101. 


93.  Sine  and  Tangent  Laws.  We  shall  require  to 
use  these  sine  and  tangent  formulae  frequently.  It  is  desirable 
that  the  student  should  realise  that  they  depend  merely  on 
mechanical  principles,  and  do  not,  except  in  the  expression 
for  R  in  terms  of  M  and  r,  involve  any  magnetic  theory. 

Consider  a  rod  AGE  (Fig.  102)  pivoted  at  its  centre  so 
that  it  can  turn  in  a  vertical  plane,  and  let  two  strings  be 
attached  to  it  at  A.  Let  one  of  these  strings  carry  a 


92-93] 


MAGNETIC    CALCULATIONS 


149 


weight  W.  If  the  other  string  be  left  free,  the  rod  will  hang 
in  a  vertical  position  under  the  pull,  due  to  the  weight  W, 
and  we  will  call  this  the  controlling  force.  Now  apply  a  force 


P  to  the  second  string.  The  rod  will  be  deflected  by  this 
deflecting  force,  and  if  a  graduated  circle  be  attached  to  a 
board  behind  the  rod  with  its  centre  at  (7,  the  angle  of 
deflexion  can  be  measured.  This  angle  of  deflexion  which  we 
will  call  </>,  will  depend  on  the  magnitude  and  direction  of  P, 
and  we  can  find  a  relation  between  P,  W,  <f>  and  if/,  the  angle 
between  P  and  W,  which  will  be  exactly  the  same  as  that 
given  by  the  equations  of  the  preceding  Section. 

Let  the  string  AE,  by  which  the  force  P  is  applied,  pass 
over  a  pulley  E,  and  carry  a  small  scale-pan  into  which 
weights  can  be  put.  The  force  P  will  be  measured  by  these 
weights,  including  of  course  that  of  the  scale-pan.  By 
making  the  distance  AE  large  it  can  be  arranged  that  AE 
is  practically  horizontal  for  all  positions  of  the  magnet,  but  in 
a  more  complete  form  of  the  apparatus  the  position  of  E  can 
be  adjusted  to  secure  this.  By  varying  the  position  of  E,  the 
direction  of  the  force  can  be  varied.  Prof.  Ayr  ton  has  devised 
several  pieces  of  apparatus  for  readily  doing  this ;  one  simple 
plan  due  to  him  is  shewn  in  Fig.  103. 


150 


MAGNETISM 


[CH.  X 


EXPERIMENT  20.     To  verify  the  tangent  laiv. 


In  this  case  it  is  necessary  that  the  deflecting  force  should 
be  at  right  angles  to  the  controlling  force,  i.e.  AE  must  be 
horizontal. 


Fig.  103. 

Let  the  weight  W  be  500  grams,  and  place  1 00  grams  in 
the  scale-pan.  Adjust  the  pulley  until  the  string  AE  is 
horizontal.  This  can  be  secured  by  observing  when  it  is 
parallel  to  a  horizontal  line  marked  on  the  apparatus,  and 
when  this  is  the  case  observe  the  angle  of  deflexion  <£,  by 
means  of  the  pointer  attached  to  the  rod.  Now  change  P, 
making  it  say  200  grams,  and  again  adjusting  the  pulley 
observe  a  new  value  for  <J>. 

Proceeding  in  this  way,  make  a  table  of  the  corresponding 
values  of  P/W  and  tan<£.  It  will  be  found  that  the  two 
quantities  are  equal. 

Thus  when  the  deflecting  force  is  at  right  angles  to  the 
controlling  force,  the  ratio  of  the  two  is  equal  to  the  tangent 
of  the  angle  of  deflexion. 


93]  MAGNETIC   CALCULATIONS  151 

EXPERIMENT  21.      To  verify  the  sine  law. 

In  this  case  it  is  necessary  that  the  deflecting  force  should 
be  perpendicular  to  the  rod. 

Proceed  as  in  the  last  experiment,  but  adjust  the  position 
of  E  until  the  line  EA  is  at  right  angles  to  the  pointer, 
determining  when  this  is  the  case  by  aid  of  a  set-square,  and 
thus  obtain  a  series  of  values  of  P  and  <£. 

In  this  case  we  shall  fine1  that  the  values  of  Pj  TFand  sin  <£ 
are  equal. 

Thus  when  the  deflecting  force  is  perpendicular  to  the 
deflected  bar,  the  ratio  of  the  deflecting  to  the  controlling 
force  is  equal  to  the  sine  of  the  angle  of  deflexion. 

The  magnetic  formulae  just  proved  can  be  submitted  to- 
a  direct  experimental  verification  by  the  aid  of  a  simple  piece 
of  apparatus. 

A  small  magnet  is  mounted  on  a  pivot  in  a  box  with  a 
glass  cover,  and  carries  a  long  light  pointer,  the  position  of 
which  can  be  read  on  a  horizontal  circular  scale  with  its 
centre  at  the  pivot.  The  length  of  the  pointer  is  usually 
perpendicular  to  the  magnet.  The  scale  is  mounted  on  a  long 


I    l    I    I    I    I    I  -M    I    i    i  :  *— <W=-»-  i    I    I    i    I    l    I    |    I    I 


Fig.  104. 

narrow  board  (Fig.  104).  This  rests  on  le veiling-screws,  and 
a  V-groove  is  cut  in  the  board  in  such  a  way  as  to  pass 
through  the  centre  of  the  circle.  A  straight  scale  of  centi- 
metres, the  zero  of  which  coincides  with  the  pivot,  is  fixed 
parallel  to  the  groove,  and  the  position  of  the  centre  of  a 
small  magnet  placed  in  the  groove  can  be  read  off  on  the  scale. 
It  is  convenient  that  the  graduations  on  the  circle  should 
be  so  arranged  that  when  the  axis  of  the  magnet  is  at  right 
angles  to  the  groove,  the  pointer  reads  zero  on  the  circle.  Such 
an  apparatus  constitutes  a  simple  form  of  magnetometer.  It 


152  MAGNETISM  [CH.  X 

can  be  used  in  various  ways  ;  thus  we  can  employ  it  to  verify 
the  tangent  formula.     This  as  we  have  seen  can  be  written 

2M 


where   M  is  the  moment  of   the  deflecting  magnet,.  H  the 
strength  of  the  controlling  field. 

We  may  put  this  result  in  the  form 


Now  M\H  is  constant  so  long  as  the  moment  of  the  de- 
flecting magnet  and  the  strength  of  the  controlling  field  are 
.unchanged  ;  if  then  we  measure  r  and  <£,  and  the  formula  is 
true,  this  value  of  Jr3  tan  <£  ought  to  be  constant. 

The  sine  formula  can  be  written 


which    shews    that   under  the  proper  conditions   £  r3  sin  <£  is 
constant. 

EXPERIMENT  22.  To  prove  that  in  the  tangent  position 
^  r3  tan  <£  is  constant,  where  0  is  the  deflexion  produced  by  a 
deflecting  magnet  in  the  end-on  position  with  its  centre  at  a 
distance  r  from  the  deflected  magnet  or  compass-needle. 

Remove  the  deflecting  magnet  to  a  distance.  Place  the 
magnetometer  (Fig.  105)  with  its  V-groove  east  and  west.  If 


Fig.  105. 

the  pointer  has  been  adjusted  to  be  at  right  angles  to  the  axis 
of  the  magnet,  this  is  done  by  setting  the  pointer  parallel  to 
the  scale. 


93]  MAGNETIC    CALCULATIONS  153 

Read  the  position  of  the  pointer;  if  the  circle  has  been 
adjusted  as  described  in  the  last  experiment,  the  pointer  will 
read  zero. 

Place  the  deflecting  magnet  in  the  groove  at  some  con- 
venient distance  from  the  deflected  magnet,  and  read  on  the 
scales  the  distance  between  the  centres  r  and  the  angle  of 
deflexion  fa. 

Reverse  the  deflecting  magnet,  replacing  it  in  the  groove 
with  its  centre  in  the  same  position  as  previously,  but  with 
its  north  pole  where  its  south  pole  was. 

The  needle  will  be  deflected,  but  in  a  direction  opposite 
to  that  of  its  previous  motion.  Read  the  deflexion  fa. 

If  everything  be  perfectly  adjusted,  fa  will  be  equal  to  fa ; 
if  the  two  do  not  differ  greatly,  the  mean  J  (fa  +  fa),  which  we 
will  call  fa  will  be  free  from  errors  due  to  the  fact  that  the 
pointer  is  not  accurately  at  right  angles  to  the  deflected 
magnet,  and  that  the  magnetic  centre  of  the  deflected  magnet 
may  not  be  midway  between  its  ends. 

K  great  accuracy  is  wanted  remove  the  deflecting  magnet  from  the 
groove  and  replace  it  on  the  other  side  of  the  needle,  but  at  the  same 
nominal  distance  from  it  as  previously,  and  read  the  deflexions  0/  and 
02'  as  before.  The  mean  of  the  four  |  (fa  +  <A2  +  <£/ +  0./)  will  eliminate 
errors  which  may  arise  from  the  centre  of  the  needle  not  being  exactly 
over  the  centre  of  the  scale. 

Now  remove  the  deflecting  magnet  to  another  distance  r2, 
and  repeat  the  observations.  Do  this  for  five  or  six  distances, 
and  then  make  a  table,  containing  in  consecutive  columns  the 
values  of  r,  fa,  fa,  <£  [=  J  (fa  +  fa)],  ?-'',  tan  <£  and  |r3  tan  fa 

It  will  be  found  that  the  numbers  in  the  last  column  are 
approximately  constant. 

We  thus  verify  the  result  that  for  a  given  deflecting 
magnet  and  control  field  the  quantity  |  r3  tan  <£  is  constant, 
and  the  theory  shews  us  that  this  constant  is  the  ratio  of 
the  moment  of  the  magnet  to  the  strength  of  the  field. 

It  may  easily  happen  that  while  the  deflected  magnet  is  so  small  that 
our  approximate  formulae  will  hold  for  it,  the  length  of  the  deflecting 
magnet  can  not  be  treated  as  very  small  compared  with  the  distance  r. 

In  this  case  we  must  have  recourse  to  a  more  complete  formula. 


154  MAGNETISM  [CH.  X 

The  fundamental  result  RIH=t&n(f>  holds,  but  the  formula  for  E  is 
less  simple  than  we  have  assumed. 

Let  21  be  the  distance  between  the  poles  of  the  deflecting  magnet  —  21 
will  really  be  less  than  the  length  of  the  magnet,  and  cannot  be  deter- 
mined with  any  accuracy,  but  if  we  treat  the  magnet  as  solenoidal,  we 
may  use  for  21  the  length  of  the  magnet,  and  we  shall  then  introduce 
rather  too  large  a  correction. 

Then  we  have  seen,  Section  91,  that  the  value  for  R  is 


_2Mr 


Hence  instead  of  the  formula 


M     1  (r2  -  f2)2 
we  have  —  =  - tan  </>, 

and  if  we  are  working  to  a  degree  of  accuracy  which  does  not  permit  of 
neglecting  the  ratio  Z2/r2,  we  should  verify  that  £  (r2  -  J2)2  tan  0/r  is 
constant. 

We  observe  that  the  value  of  the  constant  is  M/H,  so  that  if  we  know 
M  we  can  use  the  observation  to  find  H  or  conversely  if  we  know  H  we 
can  find  M. 

EXPERIMENT  23.  To  prove  that  in  the  sine  position  the  ratio 
|rssm</>  is  constant,  where  <f>  is  the  angle  of  deflexion  produced 
by  a  deflecting  magnet  with  its  centre  at  a  distance  r  from  a 
deflected  magnet  or  small  compass-needle. 

We  can  use  the  magnetometer  for  this  experiment  also. 
Place  the  deflecting  magnet  in  position,  and  then  turn  the 
whole  instrument  (Fig.  106)  round  until  the  pointer  on  the 
circle  reads  zero,  i.e.  is  parallel  to  the  groove.  The  groove  of 
course  no  longer  points  east  and  west.  Now  remove  the 
deflecting  magnet  to  a  distance.  The  needle  will  move  and 
come  to  rest  with  its  axis  north  and  south.  Read  the  pointer. 
The  reading  fa  will  be  the  deflexion  of  the  needle  produced  by 
the  deflecting  magnet  in  the  given  position. 

Reverse  the  deflecting  magnet  and  proceed  as  above  to 
find  fa.  Then  form  a  table  giving  values  of  r,  fa,  fa, 
<f)  =  |  (fa  +  fa\  r3,  sin  <f>  and  ^r3  sin  </>. 

The  last  series  of  numbers  will  be  approximately  constant, 


93-94] 


MAGNETIC    CALCULATIONS 


155 


and  if  the  same  deflecting  magnet  is  used  as  in  the  tangent 
law  experiments,  the  value  of  the  constant  MjH  will  be  the 
same  as  for  those  experiments. 


Fig.  106. 


If  21  the  length  of  the  deflecting  magnet  is  too  long  to  be  neglected, 
we  have  to  replace,  as  in  Experiment  23,  the  quantity  r3  by  (r2  -  r2)2/?-. 

The  formulae  obtained  for  the  broadside-on  position  may 
be  verified  in  the  same  way,  but  the  apparatus  is  not  quite  so 
convenient  for  this,  for  the  deflecting  magnet  has  to  be  placed 
with  its  length  at  right  angles  to,  instead  of  parallel  to,  the 
groove,  and  the  groove  will  run  north  and  south. 

94.  Law  of  the  Inverse  Square.  The  relation 
between  the  deflexions  in  the  end-on  and  broadside-on  positions 
requires  further  consideration,  for  if  <£  and  </>'  be  the  two 
deflexions  respectively  observed  with  the  distance  r,  the  same 
in  the  two  cases,  then 

M 

|  r3  tan  </>  =  —  -  r3  tan  <j> . 

Hence  tan  </>'  =  J  tan  <£. 

This  can  be  verified  by  observing  the  deflexion  <f>  in  the 
end-on  position,  then  placing  the  deflecting  magnet  in  the 
broadside-on  position,  and  observing  the  value  of  </>'  for  the 
same  value  of  r. 


156  MAGNETISM  [CH.  XI 

The  results  we  have  just  arrived  at  afford  a  conclusive 
proof  of  the  Law  of  the  Inverse  Square.  The  results  that 
Jr3tan<£  or  |r3sin<£  are  constant,  and  that  tan  $/tan  <£'  =  2 
have  both  been  arrived  at  on  the  assumption  that  the  law  of 
force  between  two  poles  is  that  of  the  inverse  square.  Ex- 
periment proves  these  results  true,  and  we  infer  therefore 
that  the  inverse  square  law  is  true  also.  Now  it  is  possible 
to  arrange  apparatus  to  measure  the  deflexions  with  much 
greater  accuracy  than  with  the  magnetometer  just  described, 
when  this  is  done  the  law  is  very  fully  verified. 

Gauss,  the  great  German  magnetician,  shewed  that  if  we  suppose  the 
law  of  force  between  two  poles  to  be  1/r71  instead  of  1/r2,  then  the  accurate 
tangent  formulae  for  the  two  positions  become 

tan  <f>  =  Lir-(n+V  +  Z,3r-(»+3)  +  . . . 
and  tan  0'  =  L/r-C'+i)  +  L3'r~(n+V  +..., 

where   -L1?  L^   are   numerical   coefficients  depending  on  the  magnetic 
moment,  and  on  the  controlling  force,  and  where  L^L^n. 

Now  Gauss  found  as  the  result  of  his  experiments  that  he  could 
express  tan  0  and  tan  <f>  by  the  two  series 

tan  0  =  0'086870r-3  -  0'002185r-5 
and  tan  0'  =  0-043435?-3  +  O0024497-5, 

and  these  held  for  values  of  r  from  about  1  to  4  metres. 

These  formulae  contain  a  double  verification  of  the  law.  In  the  first 
place  the  value  of  n  + 1  is  3,  or  n  is  equal  to  2. 

In  the  second  the  ratio  of  LJL^,  which  by  theory  is  n,  is 

•08G870/  -0043435, 
and  this  also  is  exactly  2. 

Thus  we  may  infer  from  these  experiments  that  to  a  very  high  degree 
of  accuracy  the  force  between  two  magnetic  poles  is  inversely  proportional 
to  the  square  of  the  distance  between  them. 


CHAPTER  XL 

MAGNETIC   MEASUREMENTS. 

95.     Experiments  with  the  Magnetometer.     We 

can  use  the  magnetometer  described  in  the  previous  sections 
for  various  other  experiments. 

EXPERIMENT  24.  To  compare  the  magnetic  moments  of  two 
mat/nets,  or  of  the  same  magnet  after  various  treatments. 

(a)     By  the  method  of  equal  distances. 

Adjust  the  magnetometer  so  that  the  groove  points  east  and 
west  ;  and  observe  the  reading  of  the  pointer.  Place  one  of  the 
two  magnets  on  the  scale,  with  its  north  end  pointing  to  the 
needle,  and  at  some  convenient  distance  away  and  observe  the 
deflexion,  let  it  be  fa.  Remove  this  magnet  to  a  distance, 
and  place  the  second  magnet  on  the  scale  taking  care  that  its 
centre  is  in  exactly  the  same  position  as  that  of  the  first.  Let 
the  deflexion  be  fa2.  Then  if  Mlt  M2  are  the  moments  of  the 
two  magnets,  we  have 


and  hence  Ml  :  M2  =  tan  fa  :  tan  fa. 

Thus  the  magnetic  moments  are  compared. 

The  value  to  be  chosen  for  r  —  the  "convenient  distance"  —  depends 
on  the  length  and  strength  of  the  magnet  ;  it  should  be  as  large  as 
possible,  consistent  with  giving  a  deflexion  which  can  be  measured  with 
sufficient  accuracy.  Suppose  for  example  that  the  circle  is  such  that  we 
can  read  the  position  of  the  needle  to  a  quarter  of  a  degree,  and  that  we 
want  to  know  M  to  1  per  cent.  Then  we  have  to  find  for  what  value  of 
0  an  error  of  0°'25  makes  an  error  which  is  less  than  '01  in  tan  0. 


158  MAGNETISM  [CH.  XI 

A  table  of  tangents  will  shew  that  this  is  the  case  if  0  is  greater  than 
25°  and  less  than  65°.  Thus  the  distance  should  be  such  that  the 
deflexion"  is  not  less  than  25°.  So  far  as  the  measurement  of  tan  0  is 
concerned,  greater  accuracy  will  be  secured  by  reducing  the  distance  so 
as  to  make  0  about  45°,  but  then  the  formula  depends  on  the  assumption 
that  2Z2/r2  may  be  neglected,  and  if  r  is  made  too  small  this  condition 
can  not  be  fulfilled.  It  may  of  course  happen  that  if  the  magnet  is  weak 
the  value  of  r  required  to  give  a  deflexion  of  even  25°  is  too  small  to 
permit  of  the  neglect  of  2Z2//-2  and  in  this  case  the  required  accuracy  can 
not  be  attained ;  we  must  have  recourse  to  some  more  delicate  method  of 
reading  the  deflexion. 

If  the  two  magnets  happen  to  be  of  the  same  length,  or  nearly  of  the 
same  length,  it  is  not  necessary  that  2T2/?-2  should  be  so  small  as  to  be 
negligible. 

For  if  Zj ,  1.2  be  the  lengths  we  have 


"  r 


Hence  ^  _  (r*  -  i,*)' tan 

~ 


and  if  /:  is  equal  to  12  this  ratio  is  tan  0x/tan  02  for  any  value  of  r. 
If  (Z/r)4  can  be  neglected  while  (Z/r)2  must  be  retained  we  have 

W2  ~  j          "r2       [   tan^2 » 
and  the  formula  is  still  true  if  2  (l^  -  l%2)lr-  is  sufficiently  small. 

It  is  clear  that  the  accuracy  of  the  result  is  increased  by  taking  the 
four  readings  of  0  as  described  in  Experiment  22,  i.e.  by  placing  the  centre 
of  the  magnet  first  to  the  east  then  to  the  west  of  the  needle  and  in  each 
of  these  positions  observing  (i)  with  the  north  pole  pointing  east, 
(ii)  with  the  north  pole  pointing  west. 

(b)     By  the  method  of  equal  deflexions. 

Place  the  first  magnet  in  the  end-on  position  at  a  distance 
r^  from  the  needle  so  as  to  give  a  convenient  deflexion  <£. 
Remove  the  first  magnet,  and  place  the  second  with  its  centre 
at  a  distance  ?*2,  adjusting  this  until  the  deflexion  is  again  <f>. 

Then  M1  =  |  Hr^  tan  </>, 

J/,  =  l#r98tan</>. 


Hence  Ml :  M2  =  rx3 :  r23. 

The  remarks  made  under  (a)  as  to  the  choice  of  values  of 
and  <£  apply  here.     The  experiment  may   be  modified   by 


95]  MAGNETIC   MEASUREMENTS  159 

placing  the  first  magnet  in  position  at  a  distance  rl  so  as  to 
give  a  deflexion  <£,  and  then  placing  the  second  magnet  on  the 
opposite  side  of  the  needle  at  a  distance  ra  so  as  to  bring  the 
pointer  back  to  zero. 

The  forces  on  the  needle  due  to  the  two  magnets  are  thus 
equal  and  opposite,  one  produces  a  deflexion  <£,  the  other  a 
deflexion  -  <.  Hence  we  have 


2 
T  =  iorce  due  to  first  magnet  =  —  ~  . 

>•    °  A*    « 

M  *2 

Hence  ^  :  l/2  =  r^  :  r.23. 

EXPERIMENT  25.  To  measure  the  magnetic  moment  of  a  mag- 
net having  given  a  magnetometer  in  a  field  of  known  strength, 

Place  the  magnet  with  its  axis  east  and  west,  and  in  the 
end-on  position  at  a  known  distance  r  from  the  needle  of  the 
magnetometer,  and  observe  the  deflexion  </>. 

Then,  Proposition  10,  Section  92,   M=  Jr3  //  tan  </>, 

and  if  H  is  known,  M  can  be  found.  As  before  greater 
accuracy  is  secured  by  taking  observations  with  the  magnet  in 
the  four  positions. 

If  H  be  the  horizontal  component  of  the  earth's  field,  its 
value  in  England  is  about  -18  in  c.  G.  s.  units.  See  Section  102. 

Examples.  (1)  A  magnet  placed  ivith  its  centre  at  a  distance  of  20  cm. 
from  a  magnetometer  needle  deflects  it  35°.  Find  its  moment  assuming  the 
value  of  H  to  lie  -18  units. 

8000 
Hence  M=  -^    x  -18  x  tan  35 

=  4000  x  -18  x  -700 
=  504. 

(2)  The  lengtlt  of  the  magnet  is  4  cm.,  find  the  strength  of  either  pole 
assuming  them  to  be  at  the  ends,  and  obtain  a  more  correct  expression  for 
the  magnetic  moment,  allowing  for  the  fact  that  Z2/r2  is  not  very  small. 

For  the  strength  of  the  poles  we  have 

Magnetic  Moment      504 

Strength  =  -  —  =  -7-  =  126  units. 

Length  4 


160 


MAGNETISM 


[CH.  XI 


Or  in  other  words  each  pole  of  the  magnet  if  it  were  isolated  would 
exert  a  force  of  126  dynes  on  a  unit  pole  at  a  distance  of  1  cm.  or  of 
1'26  dynes  on  a  unit  pole  at  a  distance  of  10  cm. 

To  correct  the  value  of  the  magnetic  moment  already  found  for  the 
length  of  the  magnet,  we  have  in  this  case  21  =  4,  1  —  1, 


=  -09  x  7840-8  x  -700 

=  494, 

and  the  strength  of  each  pole  is  123-5.    Thus  the  error  made  by  neglecting 
(V2/r)2  in  this  case  is  2  per  cent. 

96.  The  Mirror  Magnetometer.  More  accurate 
results  can  be  obtained  in  many  of  these  experiments  by  the 
use  of  a  mirror  magnetometer.  If  we  have  to  use  a  pointer 
made  of  aluminium,  or  of  a  fibre  of  glass,  or  some  other  such 
material,  the  graduated  circle  on  which  the  deflexions  are  read 
cannot  well  be  more  than  10  to  20  cm.  in  diameter,  and  be- 
sides difficulties  are  introduced  by  the  friction  at  the  pivot 
which  carries  the  magnet. 

In  the  mirror  magnetometer  (Fig.  107)  the  magnet  is  at- 
tached to  the  back  of  a  small  mirror  which  is  suspended  from 
a  suitable  support  by  a  fine  silk  or 
quartz  fibre  ;  rays  of  light  from  a  slit 
in  front  of  a  lamp  fall  on  the  mirror, 
and  are  reflected  on  to  a  scale,  placed 
at  right  angles  to  the  line  joining  the 
slit  and  mirror,  and  form  there  an 
image  of  the  slit.  As  the  mirror  moves, 
this  image  moves  on  the  scale,  and 
since  the  distance  between  the  mirror 
and  scale  may  be  considerable,  say 
from  one  to  two  metres,  a  very  small 
angular  motion  of  the  mirror  produces 
a  considerable  motion  of  the  spot  on 
the  scale.  The  pointer  attached  to 
the  magnet  is  in  this  case  virtually 
the  beam  of  light  and  as  this  may  easily  be  from  20  to  100 
times  as  long  as  any  possible  material  pointer,  the  sensitiveness 
is  greatly  increased. 


Fig.  107. 


95-96] 


MAGNETIC    MEASUREMENTS 


161 


The  mirror  used  in  such  a  magnetometer  may  either  be 
plane  or  concave ;  if  a  plane  mirror  is  used  a  convex  lens  of 
suitable  focal  length  is  necessary  in  order  to  form  the  real  image. 
The  lens,  which  is  shewn  at  L  in  Fig.  108,  is  placed  between 


Fig.  108. 

the  slit  S  and  the  mirror  M  in  such  a  position  that  S'  the  real 
image  of  the  slit  which  would  be  formed  by  the  lens  if  the 
mirror  were  removed  would  be  as  far  behind  the  mirror,  as  the 
scale  is  in  front  of  it.  The  light  reflected  from  the  mirror 
passes  by  the  lens  and  forms  at  Si  a  reflected  image  of  S'. 
This  is  a  real  image  of  the  slit. 

Sometimes  the  lens  is  placed  close  to  the  mirror  so  that 
the  reflected  as  well  as  the  incident  light  passes  through  it. 
In  this  case,  Fig.  109,  S  is  at  the  principal  focus  of  the 
lens.  The  rays  from  S  after  traversing  the  lens  fall  as  a 
parallel  pencil  on  the  mirror,  and  are  reflected  as  a  parallel 
pencil ;  this  is  brought  to  a  focus  at  Si  on  the  scale,  which 
is  approximately  at  the  same  distance  from  the  lens  as  the 
slit  S. 


Fig.  109.  Fig.  110. 

If  a  concave  mirror  is  used  the  slit  is  placed  at  the  same 

G    E.  11 


162  MAGNETISM  [CH.  XI 

distance  from  the  mirror  as  its  centre  of  curvature,  Fig.  110  ; 
the  rays  falling  on  it  are  reflected  to  Sl}  forming  a  real  image 
at  the  same  distance  from  the  mirror  as  the  slit. 

It  remains  to  consider  how  the  displacement  of  the  spot  on 
the  scale  is  connected  with  the  angular  motion  of  the  mirror 
and  magnet. 

Let  S  be  the  slit,  Fig.   Ill,  and  £,.  its  reflected  image, 
suppose  that  in  the  undisturbed  condi- 
tion the  adjustments   are  such   that  S 
and  $!  coincide,  or  rather  that  S,  is  just 
vertically  above  S.     When  the  magnet 
is  disturbed  let  MN  be  the  direction  of 
the  normal  to  the  mirror;  then  originally          i 
N  coincided  with  S,   and  NMS  is  the 
angle    through   which   the    magnet   has 

been  deflected,  let  this  be  <£  ;  let  SS1  the  displacement  on 
the  ssale  be  a  centimetres,  and  let  8M  be  d  centimetres. 
Then  since  the  ray  SM  is  reflected  along  MS-^  we  have 


Hence  L  SMS,  =  L  2SMN  =  2<j>, 

or  the  spot  of   light   is    deflected    through    twice   the   angle 
through  which  the  magnet  is  turned. 

Moreover,  since  S,SM  is  a  right  angle,  we  have 


Hence  a  =  d  tan  2<£. 

If  then  we  know  d  and  observe  a,  the  displacement  of  the  spot, 
we  can  find  the  angle  2<£  from  a  Table  of  Tangents,  and 
hence  we  can  obtain  <£  the  deflexion  of  the  magnet. 

In  practice  when  this  method  is  used  0  is  small,  and  we  know  that 
the  circular  measure  of  a  small  angle  is  approximately  equal  to  its 
tangent.  Hence  if  0  is  required  in  degrees,  instead  of  writing  a/d  =  tan  20 
we  may  write 

-j  =  2  x  circular  measure  of  0. 

1  a  180° 
Hence  *=--          . 


96-97]  MAGNETIC    MEASUREMENTS  163 

However,  we  usually  require  to  know  not  0  but  sin  0  or  tan  0  and  in 
the  case  where  0  is  small  we  have  as  an  approximate  result,  sufficient  for 
our  purpose, 

sin  0  —  tan  0  =  -tan20  =  -  -. 

2  '2  (t 

Example.  The  slit  is  at  a  distance  of  105  cm.  from  the  mirror,  and 
the  displacement  is  5-5  cm.  Find  the  angle  through  which  the  mirror  is 
deflected. 

Here  tan  20=  **  =  ^=-0524. 

Thus  20  =  3°  and  0  =  1°  30'. 

According  to  the  approximate  formula 


=  l°30'12"-6. 

Thus  the  difference  is  12"-6  which  is  so  small  that  for  our  purposes  it 
may  be  neglected. 

In  some  other  arrangements  a  horizontal  scale  is  placed 
before  the  mirror  and  a  telescope  is  adjusted  so  as  to  view  the 
image  of  the  scale  reflected  from  the  mirror  ;  as  the  magnet 
moves,  the  image  of  the  scale  seen  in  the  telescope  moves  also. 

The  experiments  already  described  can  be  repeated  with 
this  more  delicate  apparatus. 

97.  Measurement  of  the  Strength  of  a  Uniform 
Field  and  of  a  Magnetic  Moment.  We  have  already 
seen  how  to  determine  the  quantity  MjH,  the  ratio  of  a 
magnetic  moment  of  a  magnet  to  the  strength  of  the  field  in 
which  it  is  placed  so  that  if  we  know  the  value  of  the  moment 
we  can  find  the  strength  of  the  field  and  conversely. 

We  shall  now  see  how  we  can  find  the  value  of  MH  the 
product  of  the  magnetic  moment  and  the  strength  of  the  field 
and  hence  if  both  MH  and  MjH  are  known  we  can  find  M 


If  we  take  a  magnet  and  suspend  it  by  a  fine  fibre  so  that 
its  axis  hangs  in  a  horizontal  position  in  a  field  of  strength  H, 
it  will  oscillate  about  its  position  of  equilibrium  in  which,  if 
the  field  be  due  to  the  earth,  its  axis  would  point  north  and 
south. 

11—2 


164  MAGNETISM  [CH.  XI 

If  the  axis  be  displaced  through  a  small  angle  0  from  its 
equilibrium  position  the  couple  tending  to  bring  the  magnet  back 
will  depend  on  M.  H  being  equal  to  M.  11  sin  6.  If  we  can  find 
this  couple  experimentally  we  can  obtain  MH.  Now  we  know 
that  if  6  is  small,  sin  6  is  very  approximately  equal  to  0  so 
that  the  couple  may  be  written  MH .  6,  thus  the  couple  is  pro- 
portional to  0  the  displacement  from  rest.  In  a  case  such  as 
this  the  magnet  oscillates  backwards  and  forwards  and  it  can 
be  shewn,  both  by  theory  and  experiment,  that  the  time  of  an 
oscillation  is  a  constant1.  The  time  will  depend  on  the  shape 
and  mass  of  the  magnet,  being  greater  if  the  magnet  is  big 
and  heavy,  than  it  is  if  the  magnet  is  light.  It  will  also 
depend  on  the  restoring  couple  being  less  when  this  is  big, 
than  when  it  is  small. 

We  can  shew  from  some  dynamical  reasoning  that  the  time 
of  swing  T  is  given  by  the  formula 


where  K  is  a  quantity  called  the  moment  of  inertia  of  the 
magnet  and  depends  on  its  form  and  mass. 


From  this  we  find 


If  then  we  can  calculate  K  and  determine  T  by  experiment, 
this  equation  gives  us  the  value  of  MH. 

Now  K  can  be  found  by  measurement.  If  the  magnet  be 
a  circular  cylinder  of  mass  m  grammes,  length  21  and  radius 
a  centimetres, 


If  it  be  rectangular,  21  being  the  length,  2a  the  breadth  in  a 
horizontal  direction,  then 


Glazebrook's  Dynamics,  §  146. 


97] 


MAGNETIC   MEASUREMENTS 


165 


EXPERIMENT  26.  Having  given  a  magnet  of  known  moment 
of  inertia,  to  find  the  value  of  MH. 

Let  K  be  the  known  moment  of  inertia.  Then  MH  is 
given  by  the  formula 


where  T  is  the  time  of  a  complete  oscillation.     To  observe  T 

suspend  the  magnet  with  its  axis  horizontal,  and  protect  it  with 

a  small  bell  jar  or  some  other  covering 

to  shield  it  from  draughts  —  a  convenient 

arrangement  is  shewn  in  Fig.    112,   in 

which  the  magnet  is    suspended  inside 

a  wide-mouthed    bottle,  the   bottom  of 

which  has  been  removed  —  make  a  mark 

on  the  glass  opposite  one   end    of   the 

magnet  when  at  rest,  or  on  a  sheet  of 

paper   under   the  magnet,   and  set  the 

magnet  oscillating  through  a  small  angle 

by  bringing  a  second  magnet  near,  and 

then  removing  it. 

Determine  by  means  of  a  stop  watch 
the  time  occupied  by  a  number  of  swings 
of  the  magnet.  To  do  this  start  the 
stop  watch  as  the  end  of  the  magnet 
passes  the  mark,  and  count  the  con- 

secutive transits  reckoning  the  first  as  0,  the  second  as  1 
and  so  on.  Allow  the  magnet  to  swing  for  some  time  and 
stop  the  watch  just  as  it  passes  the  mark  when  making 
the  rath  transit.  Observe  the  number  of  seconds  the  watch 
has  been  going.  By  dividing  this  by  the  number  of  transits 
we  get  the  time  between  two  transits.  By  a  complete  period 
is  meant  the  time  between  two  transits  in  the  same  direction. 
Multiply  therefore  the  observed  time  between  two  transits  by 
2  and  we  have  T  the  time  of  a  complete  period.  Substitute  in 
the  formula 

MH=±T?\T\  we  get  MH. 

The  time  during  which  the  magnet  is  allowed  to  swing 
must  depend  on  circumstances;  if  the  magnet  will  go  on 
swinging  and  we  can  count  the  number  of  transits  without 
making  a  mistake,  the  longer  it  is  the  more  accurate  the  result. 


Fig.  112. 


166  MAGNETISM  [CH.  XI 

If  the  magnet  has  a  period  of  8  or  10  seconds  it  will 
usually  be  sufficient  to  observe  some  twenty  transits  corre- 
sponding to  a  total  interval  of  from  1J  to  2  minutes. 

We  can  put  this  formula  into  another  form  which  is 
sometimes  more  convenient. 

Let  n  be  the  number  of  transits  in  1  second.  Then 
since  a  complete  period  is  the  interval  between  2  transits 
in  the  same  direction,  ?z/2  will  be  the  number  of  complete 
periods  in  1  second,  and  2/n  will  be  the  time  of  a  complete 
period. 

Hence  T=2/n 

4_2 

and  MH  =  ^K  =  -nstfK. 

To  find  n  divide  the  number  of  transits  observed  by  the 
number  of  seconds  in  which  they  have  occurred.  On  substi- 
tuting the  value  so  found  the  same  result  will  be  obtained  for 
MH  as  previously. 

EXPERIMENT  27.  To  determine  the  moment  of  a  magnet 
and  the  strength  of  the  field  in  which  it  hangs. 

Find  as  in  Experiment  26  the  value  of  MH  and  as  in 
Experiment  25  the  value  of  MjH.  On  solving  the  two 
equations  thus  obtained  we  get  M  and  H. 

The  two  equations  are 

MH=7rW£, 

-^r 

Multiplying  them  together  we  have 
J/2=|-7rVr37f.tan 
and  dividing  the  first  by  the  second, 


Hence  both  M  and  H  can  be  found. 

Example.  The  moment  of  inertia  of  a  magnet  is  380  c.o.s.  units. 
When  allowed  to  swing  freely  in  a  field  of  strength  //  twenty  transits  are 
observed  in  2  minutes  19  seconds  and  when  placed  at  a  distance  of  30cm. 


97]  MAGNETIC   MEASUREMENTS  167 

from  a  magnetometer  needle  the  deflexion  is  10°  30'.     Find  the  value  of 
M  and  of  H. 

We  have  n=  ^=  =  -144. 

Thus  MH  =  ri>ir*K=  77'4, 

^  =  i  r3  tan  B  =  2498. 

Whence  M  -  440  c.o.s.  units, 

H  =  -176  c.o.s.  units. 

EXPERIMENT  28.  To  compare  the  strengths  of  two  magnetic 
fields. 

For  this  purpose  we  make  use  of  the  formula  MH  =  vWK, 
for  it  is  clear  that  if  the  same  magnet  be  swung  in  different 
fields  the  number  of  oscillations  in  a  second  will  vary,  the 
field  strength  being  proportional  to  the  square  of  the  number 
of  oscillations  per  second. 

Thus  if  n-i,  n2  be  the  number  of  complete  oscillations 
per  second,  Hlt  H2  the  field  strengths  in  the  two  positions, 
we  have  since  M  and  K  are  the  same 


and  hence  ffl:Ha  =  nf  :  n22. 

We  must  remember  in  using  this  method  that  there  is 
a  magnetic  field  due  to  the  earth  ;  if  all  magnets  be  removed 
from  the  neighbourhood  of  the  swinging  magnet  it  will 
oscillate  in  the  earth's  field  only,  if  another  magnet  be 
brought  near  the  field  will  be  the  resultant  of  the  earth's 
field  and  that  due  to  the  second  magnet. 

EXPERIMENT  29.  To  compare  the  strength  oj  the  field  at  a 
point  on  the  axis  of  a  magnet  produced  with  that  due  to  the 
earth  and  to  find  hence  the  magnetic  moment  of  the  magnet. 

Allow  the  vibration  magnet  to  oscillate  under  the  earth's 
field  alone  and  determine  the  time  of  twenty  transits. 

Find  hence  n^  the  number  of  oscillations  in  one  second. 

Place  the  bar  magnet  with  its  axis  north  and  south  in 
such  a  position  that  its  south  pole  is  to  the  north  of  the  centre 
of  the  vibration  magnet  and  points  to  it. 


168  MAGNETISM  [CH.  XI 

Let  the  field  due  to  the  bar  magnet  at  the  centre  of  the 
vibration  magnet  be  F.  The  resultant  field  is  F  +  H  and  if 
in  this  case  nz  is  the  number  of  oscillations  per  second 

F  +  H  _n* 
H         nf' 

F      n*-n* 

Hence  ~^-  . 

H         n* 

Thus  F  is  found  if  //  is  known. 

If  the  distance  between  the  centres  of  the  two  magnets 
be  r  centimetres  and  if  r  is  considerable  compared  with  the 
length  of  the  bar  magnet,  then 


where  M  is  the  moment  of  the  bar  magnet. 
Hence  M=  J  r3  //.  ^  ~-£?  . 

If  the  bar  magnet  is  very  long  compared  with  the 
distance  x  between  its  north  pole  and  the  centre  of  the 
vibration  magnet,  then  if  m  is  the  strength  of  either  pole 
of  the  bar  magnet,  we  have  approximately 


'=!?• 


Hence  m  = 


98.     Determination  of  the  axis  of  a  magnet. 

If  a  magnet  be  suspended  in  the  earth's  field  so  that  it 
can  move  about  a  vertical  axis,  it  will  set  with  its  axis  north 
and  south. 

The  magnetic  meridian  is  the  vertical  plane  which  contains 
the  direction  of  the  earth's  force. 

If  the  direction  of  the  magnetic  meridian  at  the  place  of 
observation  be  known  we  can  from  the  above  fact  find  the 
position  of  the  axis  of  the  magnet  ;  it  is  the  direction  in  the 
magnet  which  coincides  with  the  meridian.  Generally,  how- 
ever, the  north  and  south  line  is  not  known  with  accuracy 


97-98]  MAGNETIC    MEASUREMENTS  169 

and  we  proceed  to  shew  how  to  determine  both  it  and  the 
axis  of  the  magnet. 

Let  0  (Fig.  113)  be  the  centre  of  the  magnet,  AOB  its 
axis,  CO C '  a  line,  marked  on  one  face  of  the  magnet,  passing 
through  0. 


Ca 


Fig.  113. 

Assume  for  the  present  that  the  position  of  the  north 
and  south  line  is  known. 

Lay  the  magnet  down  on  a  sheet  of  paper  so  that  this 
face  is  horizontal  while  AOB  points  north  and  south  and  make 
marks  on  the  paper  opposite  to  the  points  C  and  C'.  Let 
them  be  Cl  and  (72. 

Remove  the  magnet  and  turn  it  over  so  that  the  face 
which  was  in  contact  with  the  paper  is  uppermost,  and  the 
face  on  which  the  line  CO'  is  drawn  next  to  the  paper. 
Replace  it  so  that  AOB  is  again  in  the  magnetic  meridian. 

Make  marks  on  the  paper  under  the  new  positions  of 
C  and  C'. 

Let  them  be  C3,  (74. 

Then  since  the  axis  of  the  magnet  and  the  line  CC'  are 
both  fixed  in  the  magnet,  the  angle  between  them  is  constant. 
And  since  in  one  position  CC'  is  over  C£!z  and  in  the  other 
over  C3C4  it  is  clear  that  the  axis  of  the  magnet  and  there- 
fore the  magnetic  meridian  bisects  the  angle  between  C-f/^ 
and  <?3<74. 

We  can  use  this  result  to  find  both  the  axis  of  the  magnet 
and  the  magnetic  meridian  thus  : 

EXPERIMENT  30.  To  determine  the  axis  of  a  magnet  and 
to  find  the  magnetic  meridian  at  any  point. 


170  MAGNETISM  [CH.  XI 

The  magnet  is  supported  by  a  stirrup  from  which  it  can 
be  easily  withdrawn  and  replaced  with  the  face  which  was 
uppermost  turned  downwards.  The  stirrup  is  suspended  over 
a  sheet  of  paper  by  a  fine  silk  fibre  from  which  the  torsion  has 
been  carefully  removed.  Two  marks  C,  C'  are  made  one  at 
each  end  of  the  magnet.  Place  the  magnet  in  the  stirrup  in 
such  a  way  that  its  axis  is  horizontal  and  allow  it  to  come  to 
rest.  It  will  rest  with  its  axis  in  the  magnetic  meridian. 
Stick  a  pin1  into  the  paper  opposite  to  each  of  the  marks 
C,  C',  let  Clt  C2  be  the  position  of  the  pins.  Remove  the 
magnet  from  the  stirrup  and  replace  it  in  the  inverted  position. 
Stick  pins  (73,  (?4  into  the  paper  opposite  the  new  positions 
of  C  and  C". 

Join  Glt  C2  and  C3,  G4,  and  bisect  the  angle  between  these 
lines. 

This  bisector  gives  the  magnetic  meridian  and  the  axis  of 
the  magnet  is  that  direction  in  the  magnet  which  is  parallel  to 
the  meridian. 

1  The  pins  should  be  of  brass. 


CHAPTER   XII. 

TERRESTRIAL   MAGNETISM. 

99.  Magnetism  of  the  Earth.  The  magnetic  force 
due  to  the  earth  varies  from  point  to  point  on  its  surface  both 
in  direction  and  in  amount.  The  direction  of  the  lines  of  force 
is  not  in  general  horizontal  but  makes  an  angle  with  the 
horizontal  plane  through  the  point  of  observation  which 
depends  on  the  position  of  that  point.  If  a  piece  of  un- 
magnetised  steel  be  suspended  from  its  centre  of  gravity  it 
will  rest  in  any  position  in  which  it  is  placed ;  if  it  be 
magnetised  it  will  set  in  a  definite  position  and  the  north 
pointing  end  will  in  these  latitudes  point  downwards.  If  the 
steel  could  be  freely  suspended  accurately  from  its  centre  of 
gravity  the  direction  of  its  axis  would  give  the  direction  of 
the  earth's  field  at  the  point ;  it  is  difficult  to  do  this,  and  so 
two  instruments  are  used,  in  one  of  these  a  magnet  is  supported 
so  that  it  can  move  in  a  horizontal  plane.  The  position  of  its 
axis  when  it  comes  to  rest  gives  the  magnetic  north  and  south, 
and  a  vertical  plane  passing  through  this  is  called  the  plane 
of  the  magnetic  meridian,  the  line  in  which  this  axis  cuts  the 
earth's  surface  is  the  direction  of  the  meridian. 

In  the  other  instrument  a  magnet  can  turn  about  a 
horizontal  axis  through  its  centre  of  gravity.  The  instrument 
is  set  so  that  this  axis  is  at  right  angles  to  the  magnetic 
meridian,  the  magnet  then  moves  in  the  plane  of  the  meridian 
and  the  direction  of  its  axis  when  it  comes  to  rest  gives  the 
direction  of  the  earth's  force. 

It  is  clear  that  the  line  of  action  of  the  earth's  force  lies 


172 


MAGNETISM 


[CH.  XII 


in  the  plane  of  the  meridian  and  that  it  can  be  resolved  into 
two  components,  horizontal  and  vertical  respectively,  in  this 
plane. 

DEFINITION.  The  angle  between  the  plane  of  the  magnetic 
meridian  and  the  true  north  and  south  line — the  astronomical 
meridian — is  called  the  Declination. 

DEFINITION.  The  angle  between  the  direction  of  the  resultant 
force  and  a  horizontal  line  drawn  in  the  plane  of  the  magnetic 
meridian  is  called  the  Dip  or  Inclination. 

Let  /  be  the  intensity  of  the  earth's  field,  i  the  dip  and  8 
the  declination. 

Then  the  direction  of  /  makes  an  angle  i  (Fig.  114)  with 
the  intersection  of   the  magnetic  meridian  and  a  horizontal 
plane    at    the    point    of    observation. 
We  can   resolve   /  into   a   horizontal  H 

component  H  and  a  vertical  component 
V,  and  we  have 


V  =  I  sin  i. 

Thus  V=Htani, 

and  I—H  sec  i. 

Hence  if  we  can  determine  H  and 
i  we  can  calculate  the  vertical  com- 
ponent  and  the  total  intensity.  ' 


Fig.  114. 


100.  Measurement  of  the  Dip.  We  have  already  seen 
(Experiment  27)  how  to  determine  the  strength  of  a  horizontal 
magnetic  field  though  of  course  additional  refinements  are 
introduced  in  accurate  instruments. 

To  find  the  Dip  we  use  a  dip  circle  (Fig.  115).  A  light 
lozenge-shaped  magnet  can  turn  about  a  very  fine  horizontal 
axis  which  passes  through  its  centre  of  gravity.  This  axis 
which  in  an  accurate  instrument  rests  on  two  polished  agate 
knife-edges  passes  through  the  centre  of  a  vertical  graduated 
circle,  so  that  the  magnetic  axis  of  the  magnet  forms  a 
diameter  of  the  circle.  When  disturbed  the  magnet  moves 


99-101]  TERRESTRIAL  MAGNETISM  173 

parallel  to  the  plane  of  the  circle,  and  when  at  rest  the 
position  of  its  ends  can  be  read  off  on  the  circle.  For  this 
purpose  microscopes  are  attached  to  the  instrument.  The 
circle  can  turn  about  a  vertical  axis  and  can  thus  be  set 
in  the  plane  of  the  magnetic  meridian. 

To  use  the  instrument  the  circle  is  turned  round  a  vertical 
axis  until  the  axis  of  the  magnet  is  itself  vertical. 

When  this  is  the  case  the  plane  of  the  circle,  parallel  to 
which  the  magnet  moves,  is  at  right  angles  to  the  meridian. 


Fig.  115. 

On  turning  the  circle  then  through  a  right  angle  the  magnet 
will  swing  in  the  plane  of  the  meridian.  Allow  it  to  come  to 
rest  and  read  the  position  011  the  circle  of  either  end  of  the 
magnet ;  the  zero  of  the  circle  is  adjusted  to  be  in  the  horizontal 
plane.  Thus  if  all  the  adjustments  are  complete  the  reading 
obtained  gives  the  dip. 

If  the  axis  of  the  magnet  does  not  pass  through  the  centre 
of  the  circle  an  error  will  be  introduced.  This  is  eliminated, 
however,  by  determining  the  dip  from  both  ends  of  the  magnet 
and  taking  the  mean. 


174  MAGNETISM  [CH.  XII 

Further  precautions  are  needed  to  eliminate  other  possible 
sources  of  error,  but  these  we  cannot  go  into  here. 

101.  Measurement    of    the     Declination.      An 

approximate  method  of  doing  this  has  already  been  described 
§  98.  In  the  more  delicate  apparatus  as  used  at  Kew  and 
elsewhere  the  magnet  is  hollow.  At  one  end  it  carries  a  scale 
photographed  on  glass,  at  the  other  a  lens  whose  focal  length 
is  equal  to  that  of  the  magnet.  Light  from  any  point  on  the 
scale  then  emerges  as  a  pencil  of  parallel  rays  from  the  lens. 
The  scale  is  viewed  through  a  telescope  which  can  turn  about 
a  vertical  axis  coincident  with  the  axis  of  suspension  of  the 
magnet,  and  the  position  of  the  telescope  can  be  read  off  on  a 
horizontal  circle  whose  centre  lies  on  the  axis  of  rotation ;  the 
telescope  has  cross  wires  at  its  focus. 

A  point  on  the  photographed  scale  is  selected — this 
corresponds  to  the  point  C  of  Fig.  113  above  and  the  line 
joining  it  to  the  centre  of  the  lens  which  is  known  as  the  line 
of  collimation  of  the  magnet  corresponds  to  the  line  CC'. 

The  telescope  is  turned  until  the  selected  point  coincides 
with  the  vertical  cross  wire  and  its  position  read  on  the 
horizontal  circle.  The  magnet  is  then  dismounted  and  in- 
verted, the  stirrup  is  arranged  for  doing  this  readily  and  the 
telescope  is  moved  until  the  same  division  of  the  scale  is  again 
on  the  cross  wire.  The  position  of  the  telescope  is  read. 

The  plane  of  the  magnetic  meridian  bisects  the  angle 
between  the  two  positions  of  the  axis  of  the  telescope,  and  is 
thus  determined  relatively  to  the  circle.  The  plane  of  the 
geographical  meridian  can  be  found  by  observations  on  the 
sun,  the  instrument  is  usually  arranged  to  make  this  possible, 
and  the  angle  between  these  two  planes  is  the  declination. 

From  the  observations  of  horizontal  intensity  and  dip  we 
can  calculate  the  total  intensity  and  the  vertical  intensity. 
If  we  also  know  the  declination  we  can  calculate  the  intensity 
in  any  given  direction. 

102.  Magnetic  Survey  of  the  Earth ;  Magnetic 
Maps.     The  values  of  the  magnetic  elements  obtained  by 
experiment  are  found  to  vary  from  point  to  point  of  the  earth's 
surface  ;  moreover  the  values  found  at  any  one  place  are  found 
to   alter   slowly    with    the    time.     The    table    drawn   up    by 


100-102] 


TERRESTRIAL   MAGNETISM 


175 


Dr   Chree  gives  the  values  of  these   quantities   at   some  im- 
portant places  as  found   in  the  year  1901. 

TABLE. 

MEAN  VALUES,  for  the  years  specified,  of  the  Magnetic  Elements  at 
Observatories  whose  Publications  are  received  at  the  National  Physical 
Laboratory. 


Place 

Latitude 

Longitude 

Year 

Declination 

Inclination 

Hori- 
zontal 
Force. 
C.G.S. 

Units 

Vertical 
Force. 
C.G.S. 
Units 

59  41  N. 
56  49  N. 
55  47  N. 
55  41  N. 
53  51  N. 
53  34  N. 
53  32  N. 
52  23  N. 
52  16  N. 
52    5N. 
51  28  N. 
51  28  N. 
50  48  N. 
50    9N. 
50    5N. 
49  12  N. 
48  49  N. 
48  15  N. 
47  53  N. 
46  26  N. 
44  52  N. 
43  43  N. 

43  47  N. 

42  42  N 
41  43  N. 
40  52  N. 

40  25  N. 

40  12  N. 

38  43  N. 
35  41  N. 
31  12  N. 
23    8N. 
22  18  N. 
19  24  N. 

18  54  N. 

14  35  N. 

6  US. 

6  49  S. 
20    68. 
22  55  S. 
37  50  S. 

30  29  E. 
60  38  E. 
49    8E. 
12  34  E. 
2  28  W. 
10    3E. 
8    9E. 
13    4E. 
104  16  E. 
5  HE. 
0  19  W. 
0    0 
4  21  E. 
5    5W. 
14  25  E. 
2    5W. 
2  29  E. 
16  21  E. 
18  12  E. 
30  46  E. 
15  51  E. 
7  16  E. 

79  18  W. 

2  53  E. 
44  48  E. 
14  15  E. 

3  40  W. 

8  25  W. 

9    9W. 
139  45  E. 
121  26  E. 
82  25  W. 
114  10  E. 
99  12  E. 

72  49  E. 

120  59  E. 

106  49  E. 
39  18  E. 
57  33  E. 
43  11  \V. 
144  58  E. 

1899 
1899 
'1897 
1900 
1901 
1900 
1900 
1900 
1899 
1899 
1901 
1900 
1900 
1900 
1900 
1901 
1898 
1898 
1901 
1898 
1900 
1899 
(1899 
11900 
1898 
1898 
1900 
(1898 
1  1899 
(1900 
11901 
1900 
1897 
1899 
1900 
1900 
1895 
(1898 
11899 
(1899 
1  1900 
1898 
1898 
1899 
1900 
1898 

0  341  E. 
9  59-6  E. 
7  54-8  E. 
10  12'2  W. 
18    9'7  W. 
11  181  W. 
12  27-7  W. 
9  56-3  W. 
2    1-5  E. 
13  54-7  W. 
16  48-9  W. 
16  29  -OW. 
14  13-6  W. 
18  291  W. 
9    7'OW. 
16  56  '5  W. 
14  53-8  W. 
8  241  W. 
7  23-4  W. 
4  41-5  W. 
9  25-3  W. 
12    4-0  W. 
5  27-8  W. 
5  28'8  W. 
13  47'0  W. 
2    5'5  E. 
9  10-2  W. 
15  51-3  W. 
15  48-4  W. 
17  201  W. 
17  161  W. 
17  18  'OW. 
4  29-9  W. 
2  20-3  W. 
3    7-8  E. 
0  18'5  E. 
7  45-6  E. 
0  28-6  E. 
0  25-4  E. 
0  51-9  E. 
0  521  E. 
1  14-9  E. 
8  181  W. 
9  32-9  W. 
7  55-7  \V. 
8  201  E. 

70  38-8  N. 
70  39-7  N. 
68  34-8  N. 
68  39-0  N. 
68  45-7  N. 

67  44-0  N. 
66  33-7  N. 
70  13-7  N. 

67    9-5  N. 

67    8-5  N. 
66    9-8  N. 
66  45-2  N. 

65  42-7  N. 
64  58-3  N. 

62  30-5  N. 
60  15-9  N. 
60  11-7  N. 
74  33-5  N. 
74  32  '5  N. 
60    1'7N. 
55  50-6  N. 

59  2^3  N. 
59  19-6  N. 
57  54-8  N. 
49    2-8  N. 
45  47  '6  N. 
52  36-0  N. 
31  24'7  N. 
44  22-2  N. 
21    6-2  N. 
21  13-9  N. 
16  19'9  N. 
16  16-0  N. 
29  47-4  S. 
36  56-8  S. 
54  16  '8  S. 
13  17  "OS. 
67  22-4  S. 

16536 
•17795 
18616 
•17513 
17348 
18152 
•18095 
•18844 
•20133 
•18502 
•18451 
•18450 
•18952 
•18689 
•19947 

•19676 
•20797 
•21175 
•22033 
•22202 
•22390 
•16503 
•16512 
•22386 
•25635 

•22768 
•22805 
•23516 
•29816 
•32825 
•30948 
•36728 
•33428 
•37445 
•37448 
•37981 
•38029 
•36752 
•28966 
•23854 
•2504 
•23364 

•47078 
•50706 
•47454 
•4480 
•44638 

•44193 
•43466 
•56009 

•43804 
•43764 
•42896 
•43507 

•42140 

•42341 
•38871 
•39087 
•59744 
•59709 
•38818 
•37784 

•38506 
•38449 
•37484 
•34356 
•33747 
•4048 
•22430 
•32764 
•14451 
•14549 
•11130 
11096 
•21040 
•21785 
•33171 
•0592 
•56050 

Katharinenburg  
Kasan  

Copenhagen  
Stonyhurst  
Hamburg  
Wilhelmshaven  
Potsdam  
Irkutsk  
deBilt(  Utrecht).... 
Kew  

Greenwich  

Uccle  (Brussels)  .... 
Falmouth 

Prague 

St  Helier  (Jersey)  .  . 
Pare  St  Maur  (Paris) 
Vienna  
O'Gyalla  (Pesth)  .... 
Odessa  
Pola  .  .  . 

Nice 

Agincourt  (Toronto). 
Perpignan 

Tiflfs  
Oapodimonte  (Naples] 

Madrid  

Coimbra  
Lisbon  . 

Tokio  .  . 

Zi-ka-wei 

Havana  .  . 

Hong  Kong 

Tacubaya  
Colaba  (Bombay).... 

Manila  

Batavia    
Dar-es  salem     
Mauritius   
Rio  de  Janeiro  
Melbourne  

176  MAGNETISM  [CH.  XII 

The  magnetic  condition  of  a  country  is  best  indicated  on  a 
magnetic  map.  Fig.  116  gives  such  a  map  constructed  for 
the  year  1900  from  the  results  of  Riicker  and  Thorpe's  Survey 
of  the  British  Isles.  The  dip  is  68°  near  Swansea,  Monmouth, 
Northampton  and  Ely.  Thus  a  line  drawn  through  these 
points  will  be  a  line  of  equal  dip.  Such  a  line  is  known  as 
an  Isoclinal  Line. 

By  finding  another  series  of  points  such  as  these  which 
have  the  same  dip  and  joining  them,  a  second  isoclinal  can  be 
drawn  and  so  on.  Thus  a  whole  series  can  be  constructed  and 
from  such  a  rnap  the  value  of  the  dip  can  be  found. 

In  the  same  way  we  can  draw  a  series  of  lines  each  of 
which  passes  through  points  at  which  the  declination  is  the 
same.  Such  a  line  is  called  an  Isogonal  Line. 

Thus  along  a  line  passing  near  Hull,  Lincoln,  Northampton, 
Oxford,  Salisbury  and  Swanage,  the  declination  in  1891  was 
18°.  This  line  then  is  an  isogonal  and  a  number  of  such 
isogonals  can  be  drawn. 

In  the  same  way  a  series  of  points  at  which  the  horizontal 
component  is  constant  can  be  found,  and  from  this  the  Lines 
of  equal  horizontal  force  can  be  constructed.  The  three 
sets  of  lines  are  all  shewn  in  the  map. 

In  a  similar  manner  lines  can  be  constructed  for  the  earth. 
The  line  at  which  the  dip  is  zero,  and  the  axis  of  the  dipping 
needle  horizontal,  passes  round  the  earth  in  a  position 
approximately  coincident  with  the  equator;  the  dip  is  90°, 
and  the  needle  stands  vertical  at  the  two  magnetic  poles. 
The  north  magnetic  pole  is  approximately  in  Latitude  70°  5'  N. 
and  Longitude  96°  43'  W.  The  position  of  the  south  magnetic 
pole  is  not  known. 

In  the  case  of  the  declination  as  we  pass  to  the  west  from 
England  across  the  Atlantic  the  westerly  declination  increases 
and  then  decreases  again  gradually  until  we  come  to  near  the 
longitude  of  Lake  Superior  when  we  cross  a  line  along  which 
the  declination  is  zero.  This  is  called  an  Agonic  Line ;  to 
the  west  of  this  line  the  declination  is  easterly,  the  magnetic 
needle  points  to  the  east  of  true  north.  If  we  travel  to  the 


102] 


TERRESTRIAL   MAGNETISM 


177 


178  MAGNETISM  [CH.  XII 

east  from  England  the  westerly  declination  decreases,  becoming 
zero  along  an  agonic  line  which  traverses  Russia  from 
St  Petersburg  to  Sebastopol,  and  then  passes  down  the 
Arabian  Gulf  to  the  west  of  India.  To  the  east  of  this  line 
the  declination  is  easterly.  There  is  another  agonic  line 
forming  an  oval  enclosing  part  of  China,  Japan,  and  the  north- 
eastern part  of  Siberia.  Within  this  oval  the  declination  is 
again  westerly. 

The  horizontal  force  again  increases  from  the  value  '18  in 
England  as  we  travel  south  across  the  Atlantic,  reaching  a 
maximum  of  about  '3  rather  to  the  south  of  the  equator  and 
falling  again  as  the  south  magnetic  pole  is  approached.  In 
parts  of  India  and  Cochin  China  it  reaches  the  value  '38. 

"  As  we  travel  towards  the  north  from  England  the  force 
falls.     At  the  magnetic  poles  its  value  is  zero.  . 

103.  Secular  Variations   of  the   Earth's  Mag- 
netism.    As  we  have  already  said,  the  values  for  the  dip, 
declination  and  force  change  with  time.     Thus  in  1576  the  dip  in 
London  was  71°  50'.     It  increased  up  to  1720  when  it  reached 
a  maximum  of  74°  42'.     Since  that  time  it  has  been  decreasing 
and  its   value  at  present  at  Kew  is   67°  9'.      In   1580   the 
compass    in    London    pointed     11°  17'    east    of    north,    the 
declination  was  easterly.     This  decreased  until  1657  when  it 
pointed  true  north,  the  declination  was  zero ;  it  then  became 
westerly  increasing  up  to   a  maximum    of    24°  30'  which  it 
reached  in  1816.     It  is  now  decreasing  again  and  at  present 
has  at  Kew  the  value  of  16°  48'  W. 

The  value  of  the  horizontal  intensity  is  increasing ;  at  Kew 
it  was  -1716  in  1814,  it  is  now  -1845  C.G.S.  units. 

104.  Daily  and  Annual  Variations.     In  addition 
to  the  above  gradual  changes  a  very  slight  daily  change  can 
be  observed.     In  the  morning  the  westerly  declination  increases 
slightly  and  continues  to  increase  till  about   1  p.m.,  it  then 
decreases  somewhat  rapidly  during  the  afternoon  and  evening, 
and  more  slowly  during  the  night ;  the  decrease  becomes  more 
rapid  during  the  earl}r  hours  of  the  morning,  until  about  7a.m. 
the  declination  is  least  and  the  increase  begins  again.     During 


102-105]  TERRESTKIAL  MAGNETISM  179 

the  summer  the  amount  of  this  change  is  about  8'.  In  winter 
the  change  is  less.  There  is  also  an  annual  change  which 
seems  to  be  related  to  the  position  of  the  sun  in  its  orbit. 

105.  Magnetic  Storms.  In  addition  to  these  regular 
changes  sudden  disturbances  of  the  magnetic  elements  occur 
from  time  to  time,  and  these  are  often  of  considerable  magnitude. 
At  magnetic  observatories  instruments  are  installed  for 
recording  the  changes  which  take  place  photographically. 


Fig.  117  a. 


Fig.  117  b. 

Fig.  1 1 7  a  is  a  reproduction  of  such  a  curve  from  the 
declination  instrument  at  the  Kew  Observatory,  shewing  a 
storm  which  occurred  on  April  10th,  1902.  The  trace  for 
an  ordinary,  quiet  day  is  also  reproduced  in  Fig.  1176. 


12—2 


180  MAGNETISM  [CH.  XII 


EXAMPLES  ON  MAGNETISM. 

1.  Several  soft  iron  needles  are  floating  vertically  very  close  to  one 
another  on  small  separate  bits  of  cork  in  a  basin  of  water.     A  powerful 
magnetic  pole  is  held  above  the  group.     Describe  and  explain  the  move- 
ments that  will  take  place. 

2.  What  force  does  a  magnetic  pole  of  strength  6  units  exert  upon  a 
pole  whose  strength  is  16  units  placed  at  a  distance  of  4  cm.  away  ? 

3.  A  pole  of  strength  8  units  acts  with  a  force  of  4  dynes  upon 
another  pole  placed  at  a  distance  of  6  cm.     Find  the  strength  of  the 
latter  pole. 

4.  A  magnetic  needle  of  pole  strength  5  units  and  length  10  cm. 
is  placed  in  a  magnetic  field  of  strength  12  so  as  to  be  at  right  angles 
to  the  lines  of  force.     With  what  couple  does  the  field  act  upon  the 
needle  ? 

5.  A  needle  of  magnetic  moment  12  is  placed  in  a  magnetic  field  of 
strength  12  units  in  such  a  direction  that  its  axis  makes  an  angle  of  30° 
with  the  lines  of  force.     Find  the  couple  acting  on  the  magnet. 

6.  Three  bar  magnets,  A,  B  and  C,  have  the  same  intensity  of 
magnetisation.     A  is  10  cm.  long  and  1  sq.  cm.  in  section,  B  is  10  cm. 
long  and  2  sq.  cm.  in  section  and  C  is  20  cm.  long  and  1  sq.  cm.  in 
section.    Compare  their  magnetic  moments. 

7.  Calculate  the  magnetic  force  at  a  point  on  the  axis  of  a  bar 
magnet  100  cm.  distant  from  the  centre  of  the  magnet,  the  strength  of 
each  pole  being  100  units  and  the  length  of  the  magnet  being  4  cm. 

8.  A  magnet  whose  pole  strength  is  2000  and  length  20  cm.  is  placed 
on  a  table.    Find  the  field  produced  at  a  point  abreast  of  its  middle  point 
and  10  cm.  distant  from  it. 

9.  The  centres  of  two  small  magnets  coincide  and  their  axes  are  at 
right  angles,  the  magnetic  moment  of  the  one  being  twice  that  of  the 
other.     Shew  that  the  lines  of  force  due  to  the  combination   are,  at 
all  points  on  the  axis  of  the  second  magnet  produced,  inclined  at  45° 
to  that  axis. 

10.  The  magnetic  moment  of  a  small  magnet  is  36  C.G.S.  units. 
Find  the  magnetic  force  due  to  it  at  a  point  in  its  axis  produced,  distant 
18  cm.  from  its  centre. 

11.  A  magnet,  suspended  horizontally,    is   caused  to   oscillate   at 
two  different  places.     At  the  first  place  it  makes   100  oscillations  in 
4  minutes,  at  the  second  110  oscillations  in  4  minutes.     Compare  the 
values  of  the  horizontal  components  of  the  magnetic  force  at  the  two 
places. 


EXAMPLES   ON   MAGNETISM  181 

12.  What   force   or   forces    must   be   applied  to   a  magnet,  whose 
magnetic  moment  is  M,  to  hold  it  fixed  in  an  East  and  West  position  ? 

(fl"=-18  C.G.S.  unit.) 

13.  A  magnet  is  suspended  by  a  wire  so  as  to  rest  horizontally  in 
the  magnetic  meridian.     When  the  upper  end  of  the  wire  is  twisted 
through  90°  the  magnet  is  deflected  30°  from  the  meridian.     How  much 
further  must  the  upper  end  be  turned  to  deflect  the  magnet  90°  from  the 
meridian  ? 

14.  A  magnetic  needle  points  North  and   South.     A  bar  magnet 
pointing  East  and  West  is  placed  with  its  centre  50  cm.  East  of  the 
needle,  and  it  is  found  that  the  needle  then  points  North-East.     Find 
the  magnetic  moment  of  the  bar  magnet  given  that  £T='18. 

15.  A  magnet  turning  about  a  vertical  axis  makes  50  vibrations  per 
minute  at  a  place  where  the  dip  is  45°,  while  it  makes  60  vibrations  per 
minute  at  a  place  where  the  dip  is  30°.     Compare  the  resultant  magnetic 
forces  at  the  two  places. 

16.  A  needle  makes  15  oscillations  per  minute  in  a  certain  magnetic 
field.     How  many  will  it  make  when  re-magnetised  so  that  its  magnetic 
moment  is  half  as  great  again  as  before  ? 

17.  A  small  magnetic  needle,  when  swinging  in  the  earth's  magnetic 
field  only,  makes  8  oscillations  per  minute.    A  long  magnet  is  placed  with 
one  of  its  poles  at  a  distance  of  8  cm.  from  the  centre  of  the  suspended 
needle,  and  in  such  a  direction  that  the  lines  of  force  due  to  the  magnet 
have,  in  the  neighbourhood  of  the  small  needle,  the  same  direction  as 
those  due  to  the  earth.     In  this  position  the  needle  oscillates  12  times 
per  minute.     If  the  long  magnet  be  now  moved  parallel  to  itself  until 
its  nearest  pole  is  now  at  a   distance   of  12  cm.  from  the   centre  of 
the  needle,  calculate  the  rate  at  which  the  latter  will  now  oscillate. 

18.  A  magnet  is  placed  with  its  axis  on  the  magnetic  meridian  and 
its  South  pole  pointing  North.     It  is  found  that  there  is  a  neutral  point 
at  a  distance  of  14  cm.  from  the  South  pole  of  the  magnet.    If  the  length 
of  the  magnet  be  10  cm.  and  H=-lSc.o.  s.  unit,  find  the  strength  of  the 
poles  of  the  magnet. 

19.  The  maximum  intensity  of  permanent  magnetisation  of  a  sfceel 
bar  10  cm.  long  and  1  sq.  cm.  in  section  has  been  found  to  be  225  C.G.S. 
units.     Find  the  tangent  of  the  greatest  angle  of  deflexion  of  a  magneto- 
meter needle  which  such  a  magnet  could  cause  if  the  needle  be  30  cm. 
from  the  centre  of  the  magnet. 

(H=  -IS  C.G.S.  unit.) 


CHAPTER  XIII. 


THE  ELECTRIC  CURRENT. 

106.  Electric  Currents.     If  two  insulated  bodies  at 
different  potentials  be  connected   to  the  opposite  quadrants 
of  an  electrometer  the  needle  is  deflected  to  an  amount  de- 
pending on  the  difference  of  potential.     If  the  two  bodies  be 
connected  by  a  conductor  this  difference   of   potential   dis- 
appears.    A    charge  of  positive  electricity  passes  along  the 
conductor  from  the  body  at  high  to  that  at  low  potential  of 
just  sufficient  amount  to  equalize  the  potentials.     This  trans- 
ference of  the  charge  constitutes  an  electric  current  in  the 
conductor.     In  such  a  case  the  potentials  are  equalized  with 
great  rapidity,  the  current  is  of  very  brief  duration.     It  is 
however    possible    by  various    means   to    maintain    a  steady 
difference   of   potential  between   the  conductors,  even  when 
connected ;  and  in  this  case  the  current  in  the  wire  is  a  con- 
tinuous one,  we  can  examine  and  measure  its  effects. 

107.  The  Voltaic  Cell.     A  voltaic   cell  is  perhaps 
the  readiest  means  by  which  this  potential  difference  can  be 
maintained.    -Such  a  cell  in  its  simplest  form  consists  of   a 
plate  of  zinc  and  a  plate  of  copper,  which  dip  separately  into  a 
vessel  containing  dilute  sulphuric  acid.     The  copper  is  called 
the  positive  plate  of  the  cell,  the  zinc  is  the  negative  plate. 
If  the  copper  and  zinc  be  connected  by  copper  wires  to  the 
opposite   quadrants   of   an  electrometer  the  needle  shews  a 
difference  of  potential.     This  difference  of  potential  is  called 
the    electromotive   force   of   the   cell.     Let  us   denote  it    by 
E.     The   unit   in  which   electromotive   force  is  measured  is 


106-107]  THE    ELECTRIC    CURRENT  183 

called  a  volt,  after  Volta,  the  discoverer  of  the  cell.  We  shall 
consider  later  how  to  define  this  unit  and  how  it  is  to  be 
measured ;  meanwhile  we  must  remember  that  when  we  say 
the  electromotive  force  of  a  battery  is  E  volts  we  mean  that 
this  difference  of  potential  exists  between  the  copper  and  the 
zinc  plates,  and  hence  that  E  volts  measures  the  number  of 
units  of  work  required  to  carry  a  unit  of  positive  electricity 
from  the  copper  to  the  zinc  plate.  In  this  case  when  the 
plates  are  not  connected  together  the  cell  is  said  to  be  on  open 
circuit. 

Two  main  theories,  the  contact  theory  and  the  chemical, 
theory  respectively,  have  been  developed  to  account  for  this 
action,  these  we  shall  consider  at  a  later  stage. 

Now  connect  the  plates  by  means  of  a  conductor.  The 
potential  difference  indicated  by  the  electrometer  falls  some- 
what— the  amount  of  fall  is  dependent  on  the  nature  of  the 
conductor — but  a  potential  difference  is  maintained  and  so 
a  continuous  current  must  flow -in  the  conductor. 

We  may  illustrate  the  process  by  considering  two  reservoirs 
filled  with  water  to  different  levels.  On  opening  communica- 
tion between  the  two  the  water  flows  from  the  reservoir  at 
higher  level  to  that  at  lower  until  the  two  levels  are  equalized, 
when  the  current  stops.  If  however  water  is  being  simul- 
taneously pumped  back  from  the  lower  to  the  upper  reservoir 
a  continuous  current  will  be  maintained 

Or  again,  imagine  two  vessels  (Fig.  118),  A,  JB,  connected 
by  two  pipes  CD,  EF ;  EF  being  at  a  higher  level  than  CD. 
Let  there  be  a  tap  G  in  EF  and  a  turbine  or  water-wheel  W 
in  CD,  which  when  it  is  turned  causes  water  to  flow  from  B 
to  A. 

On  working  the  wheel  the  level  is  raised  in  the  one  vessel, 
lowered  in  the  other;  if  a  constant  force  be  applied  to  the 
wheel  this  will  go  on  until  the  pressure  due  to  the  difference 
of  level  balances  that  due  to  the  wheel;  the  flow  ceases, 
the  difference  of  level  being  thus  maintained  at  a  steady 
height.  The  wheel  is  analogous  to  the  battery,  the  vessels  A 
and  B  correspond  to  the  two  conductors  at  different  levels. 

Now  open  the  tap  G.  There  is  a  flow  along  EF  from  A 
to  B,  the  pressure  in  A  tends  to  fall,  that  in  B  to  rise,  hence 


184 


ELECTRICITY 


[CH.  XIII 


the  turbine  is  now  able  to  propel  water  from  B  to  A  along  DC 
and  a  steady  current  is  maintained.  The  levels  in  the  two 
vessels  are  not  the  same  as  they  were  when  G  was  closed,  that 
in  A  having  sunk  to  A',  that  in  B  risen  to  B'  •  since  the  flow  is 


I 


Fig.  118. 

from  A'  to  B'  the  level  of  A'  is  above  that  of  B'  -,  the  differ- 
ence between  A  and  B'  corresponds  to  the  potential  difference 
between  the  conductors  when  in  electrical  connexion;  while 
the  difference  before  the  tap  was  opened  gives  its  value  on 
open  circuit. 

There  are  various  other  forms  of  battery  besides  the 
simple  voltaic  cell  and  various  other  methods  of  producing 
a  current ;  we  shall  recur  to  these  later ;  as  a  matter  of  fact 
the  simple  cell  described  would  not  for  various  secondary 
causes  give  a  steady  current,  in  our  experiments  we  may  use 
a  Daniell  cell,  §  127,  or  preferably  a  storage  battery,  §  131. 
In  all  cases  however  the  electromotive  force  of  the  battery  E 
volts  measures  the  potential  difference  between  its  plates  when 
on  open  circuit. 

If  the  copper  and  zinc  plates  of  a  Daniell  cell  be  connected 
by. a  wire  the  current  in  the  wire  is  from  the  copper  to  the 
zinc. 


107-109]  THE   ELECTRIC   CURRENT  185 

108.  Measure  of  a  Current.  Let  us  suppose  that 
we  have  a  wire  the  ends  of  which  are  connected  with  a 
battery  whose  electromotive  force  is  E  volts  ;  a  current  is 
flowing  in  the  wire  and  we  must  proceed  to  consider  its 
measurement  and  its  effects. 

We  measure  a  uniform  current  of  water  or  other  liquid 
flowing  in  a  tube  by  the  quantity  of  liquid  which  crosses  any 
given  section  of  the  tube  in  the  unit  of  time.  In  the  same 
way  a  uniform  current  of  electricity  is  measured  by  the  number 
of  units  of  electricity  which  cross  any  section  of  the  con- 
ductor in  1  second. 

A  point  of  some  importance  should  be  noted  here.  Let 
AB  (Fig.  119)  be  a  tube  of  variable  section  through  which 
water  is  flowing,  the  tube  being  full,  and  let  P,  Q  be  two 
sections  of  the  tube ;  if  the  tube  remains  full,  since  the  water 
is  incompressible,  the  quantity  of  water  which  crosses  P  in 
any  given  time  is  equal  to  that  which  crosses  Q  in  the  same 
time ;  if  the  tube  were  full  of  air  this  would  not  necessarily 
be  the  case,  the  compression  of  the  air  between  P  and  Q 
might  vary  and  in  consequence  there  might  be  more  air 
between  P  and  Q  at  one  period  of  the  flow  than  at  another, 
if  this  were  the  case  the  current  at  P  would  not  be  always 
equal  to  that  at  Q, 


In  this  respect  the  flow  of  electricity  when  a  steady 
condition  has  been  reached  resembles  that  of  water.  Careful 
experiment  shews  that  the  current  at  P  is  always  equal  to 
that  at  Q.  Hence  the  current  in  a  conductor  is  measured  by 
the  quantity  crossing  any  section  of  the  conductor  per  second. 

1O9.  Relation  between  Current  and  Quantity 
transferred.  Now  let  the  current  be  c.  This  means  that 


186  ELECTRICITY  [CH.  XIII 

in  one  second  c  units  of  electricity  cross  any  given  section  of 
the  conductor.  Hence  in  t  seconds  the  quantity  transferred 
is  ct  units.  Thus  if  we  denote  the  quantity  transferred  by 
q  we  have 

q  =  ct. 

We  may  of  course  write  this 

'      <4    '        •-• 

Thus  to  measure  the  current,  assuming  it  uniform,  we  have 
to  measure  the  number  of  units  of  electricity  transferred  in 
time  t  seconds  and  divide  it  by  the  time ;  the  quotient  gives 
the  current. 

We  have  already  denned  the  unit  quantity  of  electricity 
as  measured  electrostatically.  If  the  quantity  q  be  measured 
in  these  units,  the  current  given  by  the  ratio  qjt  will  be  in 
electrostatic  units,  and  the  electrostatic  unit  of  current  is  that 
current  in  which  an  electrostatic  unit  of  electricity  is  trans- 
ferred across  each  section  of  the  conductor  per  second.  We 
shall  find  however  that  for  many  purposes  this  is  not  the  most 
convenient  unit  to  employ,  when  dealing  with  electric  currents 
a  much  larger  unit  of  current  is  chosen.  This  is  called  an 
ampere  and  will  be  defined  later.  It  is  sufficient  to  say 
here  that  experiment  shews  that  a  current  of  one  ampere 
conveys  about  3  x  109  electrostatic  units  of  electricity  per 
second  across  each  section  of  the  conductor  in  which  it  is 
flowing. 

The  quantity  of  electricity  conveyed  by  1  ampere  flowing 
for  1  second  will  be  the  electro-magnetic  unit  of  quantity. 
This  is  known  as  a  coulomb. 

DEFINITION.  A  coulomb  is  the  quantity  of  electricity 
conveyed  by  1  ampere  flowing  for  1  second. 

We  may  compare  this  with  the  different  units  of  length  adopted  for 
different  purposes ;  in  some  cases  it  is  convenient  to  measure  in  milli- 
metres or  even  in  thousandths  or  millionths  of  a  millimetre,  in  others  a 
kilometre  is  selected. 

11O.     Tubes    of   Force   and   Electric   Currents. 

If  we  have  two  insulated  conductors  such  as  the  plates  of 
a  condenser,  the  one  of  which  is  charged  positively  while  the 


109-110] 


THE    ELECTRIC   CURRENT 


187 


other  is  negative,  lines  of  force  pass  as  we  have  seen  from 
the  positive  to  the  negative  conductor,  and  the  electric  forces 
can  be  represented  as  arising  from  a  tension  along  the  lines  of 
force  combined  with  a  pressure  at  right  angles  to  them,  each 
line  of  force  starting  from  a  unit  positive  charge  terminates 
in  a  unit  negative  charge.  In  Fig.  120  the  distribution  of 
the  lines  of  force  due  to  a  charged  condenser  AB  is  shewn. 
Now  let  the  plates  A,  B  be  connected  by  a  conducting  wire 
CD.  The  tubes  of  force  in  the  space  occupied  by  the  wire 
cannot  exist  within  the  material  of  the  conductor.  They 
shrink  up  into  the  wire,  the  ends  which  were  on  the  condenser 
plates  A,  B  travelling  along  the  wire  until  they  meet,  and  the 
effect  of  the  tube  is  annulled ;  the  pressure  in  the  medium  is 
thus  relieved  and  the  tubes  in  the  neighbourhood  of  the  wire 
close  on  to  it :  the  unbalanced  pressure  in  the  surrounding 


Fig.  120. 

space  forces  other  tubes  up  to  the  wire  and  these  in  their  turn 
shrink  up  into  it  until  all  the  tubes  originally  existing  be- 
tween A  and  B  have  passed  into  the  wire  and  the  field  is 
annulled.  From  this  point  of  view  we  may  look  upon  the 
transient  current  in  the  wire  as  a  transference  of  tubes  of 
force  across  the  field  up  to  the  wire  within  which  they  dis- 
appear. Now  however  suppose  that  A  and  B  are  connected 
to  the  plates  of  a  battery.  The  battery  by  its  action  generates 


188  ELECTRICITY  fCH.  XIII 

tubes  of  force  as  fast  as  they  disappear  in  the  wire,  and  the 
continuous  current  consists  in  the  passage  of  these  tubes 
across  the  field,  their  ends  sliding  as  it  were  along  the  con- 
ductor until  they  are  absorbed  into  the  wire.  Viewed  in  this 
aspect  the  current  is  made  up  of  the  transference  of  positive 
electricity  in  one  direction  combined  with  the  equal  trans- 
ference of  negative  electricity  in  the  other. 

111.  Effects  due  to  an  Electric  Current.     When 
a    current  passes  through  a  conductor  various   effects   shew 
themselves.    These  may  be  classified  as  magnetic,  thermal,  and 
chemical. 

112.  Magnetic   Action  of  a  Current.     Magnetic 
force  is  exerted  in  the  neighbourhood  of  a  wire  which  carries 
a  current.    This  was  discovered  by  Oersted,  a  Danish  professor, 
in  1820. 

EXPERIMENT  31.  To  shev)  that  a  current  in  a  wire  produces 
magnetic  force. 

(a)  Connect  a  wire  to  the  two  poles  of  a  Daniell  or  other 
cell  and  hold  it  in  a  horizontal  position  above  and  parallel  to 
a  magnet  pivoted  at  its  centre  in  such  a  manner  that  the 
current  from  the  copper  to  the  zinc  pole  flows  from  south  to 
north  in  the  wire.  It  will  be  found  that  the  north  end  of  the 
magnet  is  deflected  towards  the  west. 

If  the  wire  be  held  under  the  magnet  the  deflexion  is  to 
the  east.  If  the  direction  of  the  current  is  reversed  the 
magnet  is  deflected  to  the  east  when  the  wire  is  above,  and  to 
the  west  when  it  is  below. 

If  the  wire  be  held  in  an  east  and  west  position  at  right 
angles,  that  is,  to  the  axis  of  the  magnet,  no  deflexion  is 
observed. 

(/3)  Fix  the  wire  in  a  vertical  position  and  bring  a  small 
compass-needle  near  it.  Notice  that  the  compass  always 
tends  to  set  itself  at  right  angles  to  the  line  drawn  from  its 
centre  perpendicularly  on  to  the  wire. 

(y)  Fix  the  wire  so  that  it  may  pass  at  right  angles 
through  a  sheet  of  stiff  paper  or  cardboard  supported  in  a 
horizontal  position,  and  sprinkle  iron  filings  on  the  card ;  on 
allowing  the  current  to  pass  and  tapping  the  card  it  will  be 


110-112] 


THE   ELECTRIC   CURRENT 


189 


found  that  the  iron  filings  set  themselves  in  concentric  circles 
with  their  centres  at  the  point  in  which  the  card  is  cut  by  the 
wire. 

Thus  it  follows  from  these  observations  that  there  are 
lines  of  magnetic  force  round  the  wire,  while  the  last  two 
observations  shew  that  these  lines  are  circles  in  planes  per- 
pendicular to  the  wire.  The  wire  moreover  passes  through  the 
centres  of  these  circles. 

The  direction  of  the  force  can  also  be  determined  from  the 
first  two  observations,  and  various  rules  have  been  framed  to 
express  the  law  found. 

Thus  extend  the  right  arm  in  a  horizontal  position  with 
the  palm  downwards  and  the  thumb  pointing  to  the  left. 
Imagine  now  a  current  to  be  running  down  the  arm  from  the 
shoulder  to  the  fingers  and  that  the  thumb-nail  represents 
a  north  magnetic  pole.  Twist  the  arm  round  so  that  the 
thumb  moves  upwards  at  first.  The  direction  of  motion  of 
the  thumb  gives  the  direction  of  the  magnetic  force  due  to  the 
current  in  the  arm. 


force 


Fig.  121  a. 


If  the  magnetic  pole  be  above  the  wire  the  thumb  must  be 
held  uppermost  and  the  hand  twisted  in  the  same  direction  as 


190  ELECTRICITY  [CH.  XIII 

before,  tlvg  motion  of  the  thumb  still  gives  the  direction  of  the 


^    Or  again  we  may  state  the  rule  thus  : 

Consider  a  right-handed  screw — an  ordinary  wood  screw — 
which  is  being  screwed  into  a  piece  of  wood.  If  a  current 
flow  along  the  screw  from  the  head  to  the  point,  in  the 
direction,  that  is,  in  which  the  point  of  the  screw  is  moving, 
a  north  pole  will  tend  to  move  round  the  current  in  the 
direction  in  which  the  screw  is  being  turned. 

If  the.  current  be  reversed  so  that  it  moves  from  the  point 
to  the  head,  imagine  the  screw  as  being  withdrawn  from  the 
wood ;  the  direction  in  which  it  is  turned  will  still  give  the 
direction  of  the  magnetic  force. 

Thus  we  may  state  : 

If  a  right-handed  screw  be  placed  so  that  the  direction  of 
the  current  in  a  wire  coincides  with  the  direction  of  translation 
of  the  point  of  the  screw  when  the  screw  is  turned,  a  north 
magnetic  pole  near  the  wire  will  tend  to  move  in  the  direction  of 
rotation  of  the  screw. 

This  relation  between  the  direction  of  the  current  and  that 
of  the  force  is  illustrated  in  Figs.  121  a,  b. 

Further  experiments  may  be  made  to 
shew  that  a  current  exerts  magnetic  force. 

Thus  wind  a  piece  of  insulated  wire 
into  a  long  spiral  and  place  a  steel  knitting- 
needle  in  the  coil  with  its  length  along  the 
axis  of  the  spiral. 

On  passing  a  strong  current  through  the 
wire  and  attempting  to  withdraw  the  steel 
it  will  be  found  that  it  is  pulled  into  the 
coil  and  that  it  has  become  a  magnet. 

If  the  steel  be  replaced  by  a  piece  of 
soft  iron,  when  the  current  passes  the  iron 
becomes  temporarily  a  verypowerf ul  magnet, 
but  loses  much  of  its  magnetism  again  when 
the  current  ceases  to  flow.  Fi8-  121  &- 

Such  a  magnet  is  known   as  an  electro-magnet,  and  the 


112-114]  THE   ELECTRIC    CURRENT  191 

magnets  used  in  dynamo  machines  and  electric  motors  are  of 
this  class. 

We  shall  see  later  how  to  use  the  magnetic  effect  of  a 
current  to  measure  the  current,  at  present  our  object  is  to  get 
a  qualitative  knowledge  of  the  properties  of  a  current. 

113.  Thermal  Effects  of  a  Current.     A  current 
heats  any  conductor  through  which  it  passes. 

The  action  of  an  incandescent  lamp  is  an  obvious  illustra- 
tion of  this  fact ;  the  carbon  filament  is  brought  to  a  white 
heat  by  the  current.  Electric  heaters  of  various  forms  in 
which  this  effect  is  utilized  are  not  uncommon. 

By  the  help  of  a  calorimeter  we  can  measure  the  heating 
effect  produced  by  a  current  in  a  spiral  of  insulated  wire  or 
in  a  lamp.  Thus  if  we  have  a  water  calorimeter1  we  immerse 
the  spiral  in  a  known  mass  m  of  water  and  observe  the 
temperature  of  the  water.  On  passing  the  current  the  tem- 
perature rises  and  the  product  of  the  total  rise  of  temperature 
and  the  mass  of  the  water  gives,  apart  from  losses  from  the 
surface  of  the  calorimeter  and  other  minor  corrections,  the 
amount  of  heat  produced  in  the  wire. 

Other  thermal  effects  are  due  to  the  passage  of  a  current, 
e.g.  the  junction  of  two  metals  across  which  the  current  flows 
changes  in  temperature  as  the  current  passes,  and  the  change 
depends  on  the  direction  of  the  current,  if  it  passes  in  one 
direction  the  junction  is  heated,  if  the  direction  of  the  current 
is  reversed  the  junction  is  cooled.  A  converse  fact  to  this  is 
that  a  current  can  be  produced  by  heating  the  junction  of  two 
dissimilar  metals.  These  effects  however  are  usually  small 
compared  with  the  heating  of  the  conductor. 

114.  Chemical  Action  of  a  Current.     If  a  current 
be  allowed  to  pass  through  dilute  acid  or  through  an  aqueous 
solution  of  a  metallic  salt,  it  is  found  that  the  liquid  is  de- 
composed by  the  current.     This  phenomenon  is  called  electro- 
lysis, and  the  liquids  are  known  as  electrolytes.     Many  fused 

i  Glazebrook,  Heat,  §§  38—52. 


192 


ELECTRICITY 


[CH.  XIII 


salts  are  electrolytes,  so  are  some  solids,  e.g.  iodine  of  silver, 
and  also  probably  certain  gases. 

In  dealing  with  electrolysis  certain  terms  introduced  by 
Faraday  will  be  found  useful. 

When  a  current  traverses  a  liquid  conductor  the  surfaces  at 
which  it  enters  and  leaves  the  conductor  are  called  electrodes, 
the  surface  at  which  it  enters  the  conductor  is  the  anode,  that 
at  which  it  leaves  the  conductor  is  the  kathode. 

Suppose  now  that  we  have  two  platinum  plates  immersed 
in  dilute  sulphuric  acid  and  that  a  current  is  passed  from  one 
plate  A  through  the  liquid  to  the  second 
plate  B.  Then  A  is  the  anode  and  B  the 
kathode.  Bubbles  of  gas  collect  on  the 
two  plates,  and  if  the  products  be  ex- 
amined it  will  be  found  that  the  gas  on 
the  anode  is  oxygen  while  that  on  the 
kathode  is  hydrogen.  Moreover  it  can  be 
shewn  that  the  volume  of  the  hydrogen 
collected  in  any  time  is  twice  that  of  the 
oxygen.  This  is  most  easily  done  by  the 
use  of  the  apparatus  shewn  in  Fig.  122 
known  as  a  water  voltameter.  The  elec- 
trodes are  two  pieces  of  platinum  foil  with 
which  contact  can  be  made  from  the  out- 
side by  means  of  platinum  wire  sealed 
through  the  glass.  The  vessel  is  filled 
with  slightly  acidulated  water,  the  taps  0 
and  D  being  open ;  when  the  water  has 
risen  above  the  levels  of  the  taps  they  are 
closed.  On  passing  a  current  from  A  to 
B  the  gas  from  the  anode  rises  in  the 
inverted  burette  AC,  that  from  the 
kathode  in  the  other  burette  BD. 


Fig.  122. 


The  two  gases  are  thus  kept  separate,  and  it  will  be  noticed 
that  there  is  no  apparent  decomposition  in  the  liquid  between 
the  electrodes.  The  graduations  of  the  burettes  serve  to 
measure  the  volumes  of  the  gases  given  off,  and  it  will  be 
found  that,  when  allowance  is  made  for  the  difference  of 
pressure  to  which  the  two  are  subjected  in  consequence  of  the 


114-115] 


THE    ELECTRIC   CURRENT 


193 


difference  of  level  of  the  water  surfaces  in  the  two  burettes, 
and  for  the  difference  in  the  solubility  of  the  two  gases,  the 
volume  of  the  hydrogen  is  twice  that  of  the  oxygen. 

115.     Observations  on  Electrolysis. 

EXPERIMENT  32.    To  illustrate  the  phenomena  of  electrolysis. 

In  Fig.  123  A B  represents  a  water  voltameter,  E,,F,  G, 
are  three  beakers,  E  and  F  contain  a  slightly  acid  solution 
of  copper  sulphate,  while  G  is  filled  with  silver  nitrate. 


Fig.  123. 

Two  copper  plates  are  placed  in  E,  and  two  platinum  plates 
in  both  F  and  G.  These  are  connected  up  so  that  a  current 
can  pass  through  the  water  voltameter  and  through  the 
liquids  in  the  three  beakers  in  series.  Make  the  connexions 
and  allow  the  current  to  pass  for  some  time,  say  15  minutes, 
then  examine  the  results.  If  the  electromotive  force  of  the 
battery  used  has  been  sufficient  (the  reason  for  this  proviso 
will  appear  later)  oxygen  has  been  given  off  at  A,  hydrogen 
at  B.  The  copper  anode  in  E  is  probably  black  and  scaly — 
the  extent  of  this  depends  in  a  great  measure  on  the  purity 
and  exact  condition  of  the  materials — the  copper  kathode  in 
E  is  covered  with  a  bright  coating  of  freshly  deposited  copper ; 
the  platinum  anodes  in  F  and  G  are  unchanged,  but  oxygen 


G.  E. 


13 


194  ELECTRICITY  [CH.  XIII 

gas  has  been  given  off  from  their  surfaces  during  the  experi- 
ment; the  kathodes  however  are  covered  with  copper  and  silver 
respectively.  Electrolysis  has  gone  on  in  all  four  vessels. 

The  products  of  the  electrolysis  are  known  as  ions.  The 
ion  which  appears  at  the  kathode  is  the  kation,  the  ion 
which  appears  at  the  anode  is  the  anion.  The  metal  of  the 
solution  is  in  each  case  the  kation,  and  appears  at  the 
kathode ;  in  this  respect,  as  in  some  others,  hydrogen  behaves 
as  a  metal.  If  the  anode  can  be  attacked  by  the  anion  it 
is  dissolved  into  the  solution ;  this  is  the  case  in  E,  in  which 
the  copper  sulphate  CuS04  is  decomposed  into  the  kation  Cu 
and  the  anion  SO4.  The  latter  attacks  the  copper  of  the 
anode,  which  it  dissolves,  leaving  impurities  in  the  copper 
behind  as  scale.  In  the  vessel  F  the  same  decomposition 
takes  place  and  the  copper  appears  at  the  kathode.  The  SO4 
combines  with  the  hydrogen  of  the  water  H2O  to  form  H2SO4 
(sulphuric  acid)  and  the  oxygen  is  set  free  at  the  anode. 
In  the  third  beaker  G  the  products  are,  oxygen  at  the  anode, 
and  silver  at  the  kathode. 

116.  Faraday ?s  laws  of  Electrolysis.  These  laws 
are  two  in  number,  and  connect  together  the  amount  of  the 
ions  deposited  by  a  current  in  different  electrolytes  and  the 
quantity  of  electricity  which  passes. 

LAW  I.  The  mass  of  an  electrolyte  set  free  by  the 
passage  of  a  current  of  electricity  is  directly  proportional  to 
the  quantity  of  electricity  which  has  passed  through  the 
electrolyte. 

Thus  if  ra  grammes  of  a  substance  be  deposited  by  the 
passage  of  q  units  of  electricity,  then  m  is  proportional  to  q ;  in 
other  words,  the  ratio  m/q  is  constant  for  that  substance. 

LAW  II.  If  the  same  quantity  of  electricity  passes  through 
different  electrolytes,  the  masses  of  the  different  ions  deposited 
will  be  proportional  to  the  chemical  equivalents  of  the  ions. 

Thus  if  the  same  current  pass  through  acidulated  water, 
copper  sulphate,  and  silver  nitrate  in  succession,  then  for  each 
gramme  of  hydrogen  collected  there  will  be  8  grammes  of 
oxygen,  31*6  of  copper,  and  108  of  silver. 

Moreover  it  is  a   consequence  of  the   first  law  that  the 


115-117]  THE    ELECTRIC   CURRENT  195 

mass  deposited  depends  on  the  quantity  of  electricity  which 
has  passed,  and  not  on  the  strength  of  the  current.  A  weak 
current  flowing  for  a  long  time  produces  the  same  deposit  as 
a  stronger  current  flowing  for  a  shorter  time,  provided  the 
quantity  of  electricity  transferred  is  the  same  in  the  two  cases. 
It  should  be  noted  however  that  in  consequence  of  various 
secondary  actions  the  nature  of  the  deposit  depends  in  some 
cases  on  the  rate  at  which  it  takes  place ;  if  this  be  too  great 
the  deposit  does  not  adhere  to  the  kathode. 

117.  Electro-chemical  Equivalent.  We  can  state 
the  laws  more  concisely  by  the  introduction  of  the  term 
electro-chemical  equivalent  of  a  substance. 

DEFINITION  OF  ELECTRO-CHEMICAL  EQUIVALENT.  The 
Electro-chemical  Equivalent  of  a  substance  is  the 
number  of  grammes  of  that  substance  deposited  during  the 
passage  of  a  unit  quantity  of  electricity. 

Let  the  electro-chemical  equivalent  of  a  substance  be  y, 
and  let  m  grammes  of  the  substance  be  deposited  by  the 
transference  of  q  units  of  electricity. 

Then  since  y  grammes  are  deposited  during  the  trans- 
ference of  each  unit  of  electricity  yg  grammes  are  deposited 
by  the  transference  of  q  units.  « 

Hence  m  =  yq. 

Moreover  if  the  q  units  are  transferred  by  the  passage  of 
a  uniform  current  of  strength  c  flowing  for  t  seconds,  we  have 
q  —  ct. 

Hence  m  =  yet. 

From  this  we  have 

m 

T=T«- 

If  then  we  can  measure  the  current,  the  time  during 
which  it  has  been  flowing,  and  the  mass  of  the  substance 
deposited,  we  can  find  y,  its  electro-chemical  equivalent. 

If  this  be  done  for  a  number  of  substances,  it  will  be 
found  that  the  electro-chemical  equivalents  of  the  various 
substances  are  proportional  to  their  chemical  equivalents,  and 
this  is  what  is  stated  in  Faraday's  second  law. 

13—2 


196  ELECTRICITY  \   [CH.  XIII 


It  must  be  remembered  that  in  different  £ells  of  the 
series  each  atom  of  a  given  element  may  take  the  place  of 
one  or  more  atoms  of  hydrogen.  Thus  the  atomic  weight 
of  copper  is  63*2  but  in  copper  sulphate,  CuSO4,  each 
atom  of  copper  is  equivalent  to  two  atoms  of  hydrogen, 
thus  the  chemical  equivalent  of  copper  in  copper  sulphate  is 
63-2/2  or  31'6.  Calling  the  number  of  atoms  of  hydrogen 
which  are  replaced  by  a  given  element  in  a  given  combination 
its  "  valency  "  in  that  combination,  then  the  chemical  equiva- 
lent is  the  atomic  weight  divided  by  the  valency.  Hence  if 
we  know  the  electro-chemical  equivalent  of  hydrogen,  we  can 
find  that  of  any  other  element  by  multiplying  the  chemical 
equivalent  of  that  element  by  the  electro-chemical  equivalent 
of  hydrogen. 

The  electro  chemical  equivalent  of  hydrogen  will  depend 
on  the  unit  we  choose  to  measure  our  "unit  quantity  of 
electricity." 

If  we  suppose  that  our  unit  current  is  one  ampere,  the 
unit  quantity  of  electricity  is  the  quantity  conveyed  by  one 
ampere  flowing  for  one  second,  and  it  has  been  shewn  by 
direct  experiment  that  this  quantity  deposits  '00001038 
grammes  of  hydrogen. 

From  this  we  can  get  the  electro-chemical  equivalent  of 
any  other  substance.  Some  of  these  are  given  in  the  following 
Table, 

Electro-Chemical 
Element  Chemical  Equivalent  Equivalent 

Electro-positive 

Hydrogen  k  -00001038 

Sodium  23  \  -0002388 

Silver  108  -001118 

Copper  Cupric  31'6  -0003281 

Copper  Cuprous  63-2  -0006562 

Zinc  32-5  /                 -0003370 

Lead  103-2  /                 -001071 

Electro-negative 

Oxygen  8  {  -00008286 

Nitrogen  .  4 '66  -00004849 

Iodine  127  -001314 

The  numbers  in  the  third  column  are  in  all  cases  obtained 
by  multiplying  those  in  the  second  by  the  electro-chemical 
equivalent  of  hydrogen. 


117-119]  THE   ELECTRIC   CURRENT  197 

The  elements  are  classed  as  (1)  electro-positive,  those  which 
are  carried  forward  by  the  current  and  appear  at  the  kathode, 
and  (2)  electro-negative,  those  which  are  as  it  were  left  behind, 
being  attracted  to  the  anode  at  which  the  current  enters  the 
liquid. 

The  reasons  for  the  names  will  appear  more  clearly  when 
we  deal  with  the  theory  of  electrolysis. 

Since  one  ampere  is  one-tenth  of  the  C.G.S.  unit  of  current  (see 
Section  148),  in  order  to  find  the  quantity  deposited  by  the  passage  of 
1  C.G.S.  unit  of  electricity  we  must  multiply  each  of  the  numbers  in  the 
Table  by  10. 

118.  Local  Action.     When  a  piece  of  pure  zinc  is 
immersed  in  dilute  acid  there  is  no  action  between  the  two ; 
with  ordinary  commercial  zinc  hydrogen  is  given  off  and  the 
zinc  is  dissolved.     This  is  due  to  the  fact  that  commercial 
zinc  is  impure,  containing  among  other  things  iron.     Now  the 
iron,   the  zinc,   and  the  acid   combined  form    a    small   local 
battery,  a  current  passes  from  the  iron  to  the  zinc  and  back 
to  the  iron  through  the  acid.     This  current  causes  electrolysis 
of  the  acid  and  the  liberation  of  hydrogen.    The  S04,  which  is 
also  set  free,  attacks  the  zinc,  and  forms  sulphate  of  zinc,  thus  the 
zinc  is  dissolved  and  hydrogen  produced.     When  commercial 
zinc  is  used  in  a  battery  this  action  is  set  up,  and  unless  steps 
are  taken  to  prevent  it  the  zinc  is  being  continually  dissolved, 
even  when  the  battery  is  on  open  circuit.    This  action  known  as 
"  local    action "  is  remedied  by  amalgamating  the  zinc  with 
mercury ;  the  surface  of  the  zinc  is  first  cleaned  with  acid 
and  then  a  few  drops  of  mercury  are  run  on  to  it  and  rubbed 
in  with  a  rag  or  stick.     The  mercury  and  zinc  form  a  pasty 
mass  in  which  the  particles  of  iron  or  other  impurity  float. 
When  local  action  is  set  up  the  iron  is  carried  off  by  the 
hydrogen  bubbles  as  they  rise  to  the  surface  of  the  liquid. 
Thus  the  cause  of  the  action  is  removed  and  a  clean  bright 
surface  of  zinc  amalgam  left  for  the  acid  to  act  upon. 

119.  Voltaic  Cells.     We  have  seen  that  when  a  piece 
of  zinc  and  a  piece  of  copper  are  placed  in  dilute  acid  and 
connected  by  copper  wires  to  the  opposite  quadrants  of  an 
electrometer  a  difference  of  potential,  which  we  have  called 
the  electromotive  force  of  the  battery,  is  indicated   by  the 


198  ELECTRICITY  [CH.  XIII 

electrometer,  while  if  the  plates  be  connected  by  a  wire  the 
potential  difference  falls  and  a  positive  current  passes 
through  the  wire  from  the  copper  to  the  zinc  and  back 
through  the  acid  from  the  zinc  to  the  copper.  Now  the 
passage  of  a  current  through  the  acid  electrolyses  it.  Hydrogen 
is  carried  by  the  current  to  the  copper  plate  and  deposited  on 
it,  while  the  sulphion  S04  formed  on  the  zinc  plate,  attacks 
the  zinc  forming  zinc  sulphate  ZnS04.  The  process  may  be 
represented  by  the  equation 

Zn  +  H2SO4  =  ZnS04  +  H2. 

Zinc  and  sulphuric  acid  combine  to  produce  zinc  sulphate  and 
hydrogen. 

Moreover  the  quantities  of  the  substances  involved  in 
these  changes  are  chemically  equivalent.  For  each  gramme 
of  hydrogen  set  free,  we  have  liberated  32/2  or  16  grammes 
of  sulphur,  and  16  x  4/2  or  32  grammes  of  oxygen,  forming 
sulphion,  and  these  combine  with  65/2  or  32*5  grammes  of 
zinc  to  form  16  +  32  +  32*5  or  80 '5  grammes  of  zinc  sulphate. 
This  process  continues  but  the  current  falls  off,  for  instead 
of  having  a  copper  plate  in  the  acid,  we  have  a  copper  plate 
coated  with  hydrogen ;  the  effect  of  this  hydrogen  is  to  set 
up  a  current  in  the  opposite  direction  to  that  of  the  battery, 
and  hence  the  original  current  is  reduced.  This  effect  is  said 
to  be  due  to  polarization,  and  the  battery  is  said  to  be  polarized. 

We  have  already  seen  that  a  current  in  a  wire  exerts 
magnetic  force  in  its  neighbourhood,  an  instrument  arranged 
to  make  use  of  this  magnetic  effect  to  measure  the  current  is 
called  a  galvanometer.  See  Section  149.  By  the  aid  of  a 
galvanometer  we  can  detect  this  reverse  current  due  to 
polarization  thus: 

EXPERIMENT  33.  To  shew  the  reverse  current  due  to  polari- 
zation. 

A  batter}'  of  two  or  more  Daniell  cells,  a  water  voltameter, 
consisting  of  two  platinum  plates  immersed  in  a  beaker  of 
slightly  acidulated  water,  and  a  delicate  galvanometer  are 
connected  as  shewn  diagrammatically  in  Fig.  124. 

The  handle  S  of  a  switch  is  connected  to  the  galvano- 
meter G,  the  other  terminal  of  the  galvanometer  is  connected 
to  one  plate  of  the  voltameter  V,  and  the  second  plate  of 


119] 


THE    ELECTRIC    CURRENT 


199 


the  voltameter  is  connected  to  the  battery  B.  This  point  of 
junction  is  connected  to  one  terminal  R  of  the  switch,  the 
other  terminal  T  is  joined  to  the  second  pole  of  the  battery. 


Fig.  124. 

Thus  when  the  handle  of  the  switch  connects  S  and  T  a 
circuit  is  complete  through  the  battery,  voltameter  and  galva- 
nometer. When  the  handle  is  moved  across  to  S  the  battery 
is  cut  out  of  the  circuit,  which  is  complete  through  the  volta- 
meter and  galvanometer. 

Place  the  switch  so  that  $  and  T  are  connected.  A  current 
flows  round  the  circuit,  electrolysis  goes  on  in  the  voltameter, 
gases  being  deposited  on  the  plates,  and  the  needle  of  the 
galvanometer  is  deflected.  Note  the  direction  of  the  deflexion. 
Then  reverse  the  switch  so  that  S  and  R  are  connected,  the 
battery  is  now  out  of  circuit ;  the  galvanometer  however  still 
shews  a  current,  but  in  the  opposite  direction  to  that  which 
previously  deflected  it.  The  plates  are  polarized  and  an  E.M.F. 
of  polarization  is  acting.  The  oxygen  and  hydrogen  of  the 
voltameter  in  part  recombine,  and  as  this  combination 
becomes  more  complete  the  current  dies  away  to  zero. 

This  same  process  occurs  in  the  simple  voltaic  cell  when 
it  is  producing  a  current;  in  consequence  of  the  opposing 
E.M.F.  of  polarization  the  electromotive  force  of  the  cell  falls 
rapidly  with  use  and  the  current  diminishes.  For  this  reason 
various  modifications  of  the  cell  have  been  devised  which  aim 
at  reducing  the  polarization. 

In  most  cells  the  negative  plate  is  made  of  zinc,  the 
positive  plate  varies ;  in  all  cases  however  hydrogen  tends  to 
pass  from  the  liquid  near  the  zinc  to  the  positive  plate,  and 
the  object  aimed  at  is  to  neutralize  the  consequences  of  its 
deposition.  This  result  is  achieved  by  putting  an  oxidizing 


200  ELECTRICITY  [CH.  XIII 

agent  either  into  the  liquid  or  into  the  material  of  the 
positive  plate.  The  electromotive  forces  of  the  different 
cells — measured  it  will.be  remembered  by  the  differences  of 
potential  between  their  plates  when  on  open  circuit — are 
different. 

In  some  cells  a  single  fluid  is  used,  in  others  the  cell  is 
divided  into  two  compartments  separated  by  a  plate  of  porous 
earthenware,  the  zinc  plate  is  in  one  compartment,  the  positive 
plate  in  the  other,  and  the  two  compartments  contain  different 
fluids. 

120.  Single  fluid  cells.     In  these  the  oxidizing  agent 
may  either   be   in   the   fluid    or   in   the  positive  plate.     As 
oxidizing  agents  various  substances  might  be  used,  as  nitric 
acid,  bichromate  of   potash,   black    oxide   of   manganese,    or 
peroxide  of  lead. 

Most  of  these  however  would  attack  the  copper,  and  so 
the  copper  has  to  be  replaced  by  some  substance  which  will 
resist  the  chemical  action  of  the  oxidizer.  Carbon  and 
platinum  are  such  substances  and  are  in  consequence  used 
in  various  cells ;  thus  we  might  have  a  cell  containing  zinc 
and  carbon  in  nitric  acid  for  example.  But  nitric  acid  would 
dissolve  the  zinc  on  open  circuit.  Hence  it  cannot  be  used  in 
the  same  vessel  as  the  zinc  plate ;  we  can  only  employ  it  in  a 
two-fluid  cell. 

Bichromate  of  potash  however  dissolved  in  sulphuric  acid 
is  used  in  •  the  bichromate  cell,  though  in  this  case  also  it  is 
necessary  to  provide  means  for  withdrawing  the  zinc  plate 
when  the  cell  is  out  of  action.  In  the  Leclanche"  and  dry 
cells,  binoxide  of  manganese  is  mixed  with  the  carbon  of  the 
positive  plate,  the  exciting  agent  being  chloride  of  ammonia. 

121.  The  Bichromate  Cell.     This  is  usually  made 
up  in  the  bottle  form  shewn  in  Fig.  125.     The  positive  pole 
is  composed  of  two  plates  of  hard  carbon  connected  together 
and  to  one  binding  screw.     The  negative  pole  is  a  plate  of 
zinc  which  is  connected  to  the  other  binding  screw,  and  can 
be  withdrawn  from  the  liquid  as  shewn  in  the  figure.     The 
liquid  is  a  solution  of  bichromate  of  potash  in  sulphuric  acid. 
The  electromotive  force  of  the  cell  on  open  circuit  is  about 


119-122] 


THE   ELECTRIC    CURRENT 


201 


2-1  volts.  When  the  poles  are  connected  by  a  wire  a  current 
flows  in  the  wire  from  the  carbon  to  the  zinc.  The  sulphuric 
acid  is  electrolysed  and  zinc  sulphate  formed,  but  the  hydrogen 
combines  with  some  of  the  oxygen  of  the  bichromate  and 
polarization  is  prevented.  In  consequence  of  the  proximity 
of  the  plates  in  the  acid  solution  the  internal  resistance  (§  135) 
of  this  cell  is  low,  and  it  is  possible  to  use  it  to  give  a  large 
current  through  a  suitable  external  circuit. 


Fig.  125. 


Fig.  126. 


122.  Leclanche  Cell.  In  this  cell,  which  in  its  usual 
form  is  shewn  in  Fig.  126,  the  liquid  is  salammoniac,  the 
negative  plate  is  a  rod  of  amalgamated  zinc,  and  the  positive 
plate  a  mixture  of  carbon  and  black  oxide  of  manganese. 
This  mixture  is  pounded  up  and  then  compressed  into  a  hard 
cake ;  in  the  older  form  of  cell  it  is  put  inside  a  porous  pot. 
When  in  use  the  cell  polarizes ;  a  double  chloride  of  zinc 
and  ammonia  is  formed,  and  hydrogen  and  ammonia  are  set 
free  which  collect  on  the  carbon.  If  the  cell  however  be 
only  used  for  a  short  period,  and  then  left  on  open  circuit, 
the  manganese  binoxide  gradually  gives  off  oxygen,  which 
combines  with  the  hydrogen  and  depolarizes  the  positive  plate. 
The  cell  is  thus  very  convenient  for  use  on  telegraph  circuits, 
electric  bells,  telephones,  and  the  like  where  it  is  only  wanted 


202 


ELECTRICITY 


[CH.  XIII 


for  a  short  interval  at  a  time  and  then  has  a  period  of  rest. 
It  is  easily  set  up,  is  clean,  and  requires  very  little  attention. 
Its  electromotive  force  is  about  1'35  volts. 

123.  The  Dry  Cell.    This,  which  is  shewn  in  Fig.  127, 
is  a  modification  of  the  Leclanche  arranged  to  secure  portability. 
The  active  materials  are  the  same,  but  the 

cell  is  filled  with  a  paste  of  sulphate  of  lime, 
or  some  such  material,  soaked  in  chloride  of 
ammonia.  This  pasty  mass  carries  the  current 
in  the  same  way  as  the  solution  of  the 
Leclanch^,  the  cell  however  is  more  portable, 
and  less  liable  to  accident. 

124.  Two-Fluid  Batteries.     These 
are  designed  to  counteract  the  direct  action 
of  the  depolarizer  on  the  zinc  of  the  negative 
pole. 

The  positive  plate  is  immersed  in  a  strong 
depolarizing  solution  in  a  porous  pot.     This  F-     127 

is  put  inside  another  vessel,  which   contains 
the  zinc  plate   in    dilute   acid   or  a  solution  of   sulphate  of 
zinc.     The  electrical  action  can  go  on  through  the  pores  of 
the  pot,  but  the  depolarizer  can  only 
reach    the    zinc    slowly    by    diffusion 
through    the    pores.      Sometimes    the 
zinc  and  acid  are  in  the  porous  pot, 
the  positive  plate  being  in  the  outer 
vessel,   but   this    of   course  makes  no 
difference  to  the  action. 

125.  Grove's    Cell.      This    is 
shewn  in  Fig.  128.    The  positive  plate 
consists  of  a  piece  of  platinum  foil ; 
this  is  immersed  in  strong  nitric  acid. 
The  porous  pot  is  usually  flat  in  shape, 
and  the  negative  plate  is  a  zinc  sheet 
bent  so  as  to  encompass  the  pot  closely; 
both  are  placed  in  a  porous  pot  which 
contains  sulphuric  acid;  thus  the  plates 
are  near  together  and  the  internal  re- 


Fig.  128. 


122-127] 


THE   ELECTRIC   CURRENT 


203 


sistance,  §  135,  is  low.  The  sulphuric  acid  is  decomposed, 
forming  zinc  sulphate  and  hydrogen  ;  the  latter  passing 
through  the  pot  is  oxidized  by  the  nitric  acid  which  is 
itself  reduced.  The  electromotive  force  is  about  1-95  volts. 

126.  Bunsen's  Cell.    This  cell  is  similar  to  the  Grove 
cell,   only  the  platinum,   which  is  costly,  is  replaced  by  gas 
carbon.     Its  action  and  electromotive  force  are  both  the  same 
as  that  of  Grove's  cell. 

127.  DanielPs  Cell.    In  this  cell,  Fig.  129,  the  positive 
plate  is  copper  immersed  in  sulphate  of  copper,  the  negative 
plate,  zinc  in  sulphuric  acid.     The  two 

liquids  are  kept  apart  by  a  porous  par- 
tition ;  in  the  form  shewn  in  the  figure 
the  copper  plate  is  placed  in  the  outer 
vessel  of  the  cell,  the  zinc  plate  and 
sulphuric  acid  are  .contained  in  the 
porous  pot  which  is  placed  inside  the 
copper  vessel. 

The  sulphuric  acid  is  decomposed, 
forming  zinc  sulphate  and  hydrogen, 
the  hydrogen  traverses  the  porous  pot 
and  replaces  the  copper  in  the  copper 
sulphate,  forming  sulphuric  acid  and 
copper,  and  the  copper  is  deposited  on 
the  copper  plate.  Thus  the  nature 
of  that  plate  is  not  changed  by  the 
passage  of  the  current  and  there  is  no  polarization.  In  order 
to  maintain  the  copper  sulphate  solution  of  constant  strength 
crystals  of  sulphate  of  copper  are  placed  in  the  solution,  usually 
in  a  small  tray  at  the  top  of  the  cell,  these  are  gradually 
dissolved,  thus  replacing  the  copper  deposited  on  the  copper 
plate. 

The    chemical    action    may   be   represented    by   the    two 
following  equations  : 


Fig.  129. 


Zinc  and  Sulphuric  Acid  give  Zinc  Sulphate  and  Hydrogen. 

H2  +  CuSO4  -  H2SO4  +  Cu. 
Hydrogen  and  Copper  Sulphate  give  Sulphuric  Acid  and  Copper. 


204 


ELECTRICITY 


[CH.  XIII 


Sometimes  the  cell  is  made  up  with  a  concentrated  solution 
of  zinc  sulphate  in  water,  instead  of  the  sulphuric  acid.  The 
electromotive  force  depends  on  the  solution,  and  varies  from 
about  1-18  volts  when  sulphuric  acid  diluted  with  twelve  parts 
of  water  is  used,  to  1  '07  volts  when  zinc  sulphate  is  used. 

For  solutions  of  a  given  strength,  however,  the  electro- 
motive force  of  the  cell  is  very  constant. 

128.  Standards  of  Electromotive  Force.   A  cell  of 
constant  electromotive  force  may  conveniently  be  employed  as 
a  standard  in  terms  of  which  to  measure  the  electromotive 
force  of  any  other  cells.     For  this  purpose  the  Clark  cell  has 
been  adopted  as  a  legal  standard  in  many  countries. 

129.  The  Clark  Cell.     This  may  take  various  forms. 
One  such  form  is  shewn  in  Fig.  1 30.     The  cell  is  contained  in 
a   glass   test-tube.      The   positive  pole 

is  pure  mercury,  and  communication  is 
made  with  this  by  means  of  a  platinum 
wire  which  passes  through  a  glass  tube 
into  the  mercury.  Above  the  mercury 
is  a  paste  of  mercurous  sulphate  dis- 
solved in  pure  zinc  sulphate ;  this  is 
covered  again  by  a  saturated  solution 
of  zinc  sulphate  containing  crystals  of 
zinc  sulphate  so  as  to  remain  saturated 
at  any  temperature  at  which  the  cell 
may  be  used.  An  amalgamated  rod  of 
pure  zinc  dips  into  the  zinc  sulphate 
and  forms  the  negative  pole.  The  cell 
is  closed  with  a  cork  and  sealed  with 
marine  glue.  It  is  not  to  be  used  as 
a  source  of  a  current,  for  it  will  polarize, 
but  merely  as  a  standard  of  electro- 
motive force  on  open  circuit,  or  in  such 
circumstances  that  the  current  which  can  be  formed  must  be 
infinitesimal.  The  electromotive  force  like  that  of  other  cells 
depends  on  the  temperature.  At  15°C.  its  value  is  1-434  volts1. 

1  More  recent  experiments  appear  to  shew  that  this  number  should 
be  reduced  probably  to  1-4328,  but  tbe  question  is  now  under  investiga- 
tion. 


ZINC  SULPHATC 
SOLUTION 


ZINC  SULPHATE 
CRYSTALS 


Fig.  130. 


127-131] 


THE    ELECTRIC   CURRENT 


205 


Another  form  of  Clark  cell  which  has  many  advantages  is 
shewn  in  Fig.  131.  The  positive  pole  is  mercury  contained  in 
one  of  the  two  test-tubes,  the  negative  pole  an  amalgam  of 
zinc  and  mercury.  Communication  is  made  with  the  poles  by 
means  of  platinum  wires  sealed  through  glass  tubes. 


MERCURY-  ^^=.  r-= 


Fig.  131. 

Above  the  mercury  is  the  mercurous  sulphate  paste,  above 
the  zinc  the  saturated  zinc  sulphate  solution  containing  crystals 
of  the  salt.  Communication  between  the  two  goes  on  through 
the  horizontal  tube  which  is  filled  with  zinc  sulphate.  This 
pattern,  known  as  the  H  form  of  cell,  was  devised  by  Lord 
Rayleigh.  The  materials  are  more  completely  separated  than 
in  the  other  pattern  and  the  E.M.F.  is  more  constant. 

130.  Weston  Cell.     For  some  purposes  Weston's  modi- 
fication of  the  Clark  cell  is  very  useful.     In  place  of  the  zinc 
Weston  uses  cadmium  immersed  in  cadmium  sulphate;    the 
mercurous    sulphate    paste    is   also   made   up   with    cadmium 
sulphate.     The  great  advantage  of  the  cell  is  that  its  E.M.F. 
varies  very  little  with  temperature;  its  value  is  1'018  volts  at 
temperatures  near  15°  C. 

131.  Secondary  Batteries.     We  have  already  seen 


206  ELECTRICITY  [CH.  XIII 

that  when  two  platinum  plates  are  immersed  in  dilute  acid  they 
become  polarized  on  the  passage  of  a  current,  and  will,  if 
connected  directly,  give  a  current  for  a  brief  time  during  which 
the  polarization  is  reduced.  Plante  shewed  that  if  the  platinum 
plates  be  replaced  by  lead  plates  treated  in  a  certain  manner 
by  the  passage  of  a  current  backwards  and  forwards  it  be- 
comes possible  to  store  up  in  the  cell  a  quantity  of  electricity 
and  to  utilize  the  cell  as  a  source  of  current. 

Plante's  original  process  was  modified  by  Faure.  Two 
plates  of  lead  A  and  B,  coated  with  minium  or  red  lead,  are 
placed  in  a  cell  containing  dilute  sulphuric  acid  and  a  current 
is  passed  from  A  to  B. 

The  red  lead  on  A  becomes  peroxidized,  that  on  B.  is 
reduced  to  a  lower  oxide,  and  then  finally  to  the  condition  of 
spongy  metallic  lead.  If  these  two  plates  are  now  connected 
together,  the  original  source  of  current  being  removed,  a 
reverse  current  is  produced  passing  from  B  to  A  in  the  cell 
and  from  A  to  B  in  the  wire,  and  this  goes  on  until  the 
original  condition  is  reached. 

By  continuing  the  process  of  charging  and  discharging  for 
some  time  the  amount  of  lead  taking  part  in  the  changes  is 
gradually  enlarged,  and  thus  the  capacity  of  the  cell,  as 
measured  by  the  quantity  of  electricity  it  can  hold  before  the 
hydrogen  begins  to  come  off  in  bubbles  from  the  plate  B,  is 
considerably  increased. 

In  the  more  modern  form  of  accumulator  or  storage  cell 
the  plates  take  the  form  of  a  grid  of  metallic  lead  into  the  inter- 
stices of  which  a  paste  of  red  lead  and  sulphuric  acid  is  pressed. 
The  cell  is  formed  by  the  passage  of  a  current  which  peroxi- 
dizes  the  paste  on  one  plate  and  reduces  it  on  the  other. 

When  the  cell  is  in  use  the  specific  gravity  of  the  acid 
solution  should  be  about  1*18.  The  electromotive  force  of  the 
cell  is  about  2  volts,  and  continues  at  this  value  until  the  cell 
is  almost  completely  discharged ;  as  this  stage  approaches  a 
very  rapid  fall  in  the  electromotive  force  is  observed ;  the 
discharge  should  be  stopped  when  the  E.M.F.  reaches  1*85  volts 
and  the  cell  recharged. 

In  practice  the  capacity  of  an  accumulator  is  measured  in 
ampere-hours,  an  ampere-hour  being  the  quantity  of  elec- 


131-132] 


THE   ELECTRIC   CURRENT 


207 


tricity  conveyed  by  a  current  of    1    ampere  flowing  for  one 
hour.     It  is  thus  3600  coulombs. 

Accumulators  usually  contain  a  number  of  plates  ranged 
side  by  side,  the  odd  plates  are  connected  together  to  form  one 
pole  of  the  cell,  while  the  even  plates  connected  together  form 
the  other.  The  capacity  of  the  cell  depends  on  the  size  and 
number  of  its  plates,  its  electromotive  force  is  independent  of 
this,  and  is  determined  only  by  their  nature  and  the  state  of 
their  surfaces.  Fig.  132  shews  a  form  of  secondary  cell  in 
general  use. 


Fig.  132. 

132.  Arrangement  of  Batteries  in  Series.  Con- 
sider two  batteries  A,  B  (Fig.  133).  Let  C^  Zl  be  the  copper 
and  zinc  plates  of  the  one,  6\,  Z.2  those  of  the  second ;  El  and 
E.2  being  the  electromotive  forces.  Connect  together  Z^  and  C2. 
Connect  Cl  and  the  junction  of  Z±  and  (72  by  two  wires  to 
an  electrometer;  a  potential  difference  E^  will  be  observed, 
the  potential  of  the  quadrant  in  connexion  with  Cl  being 
higher  than  that  of  the  quadrant  connected  to  the  junction  by 
an  amount  El.  Now  connect  the  junction  and  Z%  to  the 
electrometer;  the  potential  of  the  quadrant  connected  to  the 
junction  will  be  found  to  exceed  by  E^  that  of  the  quadrant 
connected  to  Z^.  Thus  the  potential  difference  between  C^ 
and  Z±  is  El  +  E*,  and  this  may  be  verified  by  connecting  Cl 
and  Z  to  the  electrometer. 


208 


ELECTRICITY 


[CH.  XIII 


Two  batteries  arranged  so  that  the  negative  pole  of  the 
one  is  connected  to  the  positive  pole  of  the  other  are  said  to 
be  connected  in  series,  and  the  electromotive  force  of  such  an 


Fig.  133. 

arrangement  is  the  sum  of  the  electromotive  forces  of  the  two 
batteries.  In  general  if  a  number  of  batteries  are  so  connected 
the  same  law  holds  ;  if  JS19  Ez  etc.  be  the  individual  electro- 
motive forces,  then 


133.    Arrangement  of  Batteries  in  Multiple  Arc 
or  Parallel.      Consider  now 
two    cells 
E.M.F.,  E. 


having  the  same 
The  potential  dif- 
ference both  between  Cl  and 
ZD  and  between  (72  and  Z^  is 
E.  If  then  d  and  <72  (Fig.  134) 
be  connected  by  one  wire  to 
one  pair  of  quadrants  of  an 
electrometer,  and  Z^  and  Zz 
be  connected  by  a  second  wire 
to  the  other  pair  of  quadrants, 
the  electrometer  will  still  in- 
dicate a  potential  difference  E. 
The  cells  are  said  to  be  con- 
nected in  parallel  or  in  multiple 


Fig.  134. 


132-133]  THE   ELECTRIC   CURRENT  209 

arc,  and  when  the  two  have  the  same  E.M.F.  this  is  also  the 
E.M.F.  of  the  combination. 

If  the  two  cells  be  not  equal  in  E.M.F.  the  stronger  cell  will 
send  a  current  round  the  circuit  and  the  problem  of  finding  the 
potential  difference  between  the  plates  is  more  complex. 


CHAPTER  XIV. 

EELATION   BETWEEN  ELECTROMOTIVE   FORCE 
AND   CURRENT. 

134.  Electromotive  Force  and  Current  in  a 
Simple  Circuit.  When  a  difference  of  potential  is  established 
between  two  points  on  a  conductor  a  current  flows  in  the  con- 
ductor ;  it  remains  for  us  to  consider  the  relation  between  the 
strength  of  this  current  and  the  difference  of  potential  or 
electromotive  force. 

Now  experiment  proves  that  for  a  conductor  composed  of 
a  single  material  in  a  given  physical  condition  the  ratio  of  the 
difference  of  potential  between  two  points  to  the  current 
flowing  between  these  points  is  a  constant.  This  constant  is 
known  as  the  Resistance  of  the  conductor.  Let  us  denote 
this  quantity  by  ft.  Let  Vlt  V2  be  the  potentials  between  the 
points  and  let  C  be  the  current  flowing  between  them.  Then 
we  have  the  result  that 


where  R,  the  resistance,  is  a  constant  quantity  depending  on 
the  shape,  material,  and  other  physical  properties  of  the  con- 
ductor, but  not  on  the  current  or  on  the  difference  of  potential. 

We  can  of  course  put  the  equation  into  the  form 
F!-  F2 
~R      ' 

As  will  be  seen  in  the  sequel  the  resistance  of  a  conductor  depends  on 
its  temperature,  and  since  the  passage  of  the  current  heats  the  conductor 
its  resistance  is  to  this  extent  dependent  on  the  current. 


134-135]      ELECTKOMOTIVE   FORCE   AND   CURRENT          211 

The  experimental  verification  of  this  result  will  come  at  a 
later  stage  (see  §  163).  For  the  present  we  will  consider  some 
further  consequences  of  it. 

It  follows  from  the  above  that  any  conductor  has  a  definite 
resistance — we  shall  see  later  how  this  may  be  measured — and 
that  if  we  know  the  resistance  we  can  calculate  the  current 
produced  by  a  given  difference  of  potential  applied  to  the  ends 
of  the  conductor,  or  conversely  the  difference  of  potential 
required  to  produce  a  given  current. 

135.  Electromotive  Force  and  Current  in  a 
Circuit  containing  a  Battery.  The  above  statement 
assumes  that  we  have  no  source  of  electromotive  force  in  the 
circuit  between  the  points  at  which  the  potentials  V1}  F2  are 
measured ;  the  conditions  correspond  to  those  which  hold  in 
the  case  of  water  flowing  down  a  sloping  tube.  The  flow  will 
depend  on  the  nature  of  the  tube  and  on  the  difference  of 
pressure  between  the  ends. 

The  circumstances  are  entirely  altered  if  in  the  tube  we 
place  a  turbine  or  wheel  tending  to  propel  the  water  in  the 
tube  either  up  or  down.  If  there  be  a  battery  or  some  other 
source  of  electromotive  force  between  the  points  at  which  the 
potentials  are  measured  then  the  E.M.F.  of  this  battery  must 
be  added  to,  or  subtracted  from,  the  potential  difference  between 
the  points  in  order  to  get  the  resultant  electromotive  force  to 
which  the  current  is  due.  Thus  consider  a  circuit  consisting 
of  a  battery  of  E.M.F.  E,  and  resistance  A'2  the  poles  of  which 
are  connected  by  a  wire  of  resistance  jRlf  The  quantity  E  it 
will  be  remembered  is  measured  by  the  difference  of  potential 
between  the  poles  of  the  battery  when  on 'open  circuit. 

Let  Vl  be  the  potential  of  the  positive  pole,  F"2  that  of  the 
negative  pole.  Then  as  we  pass  along  the  wire  from  the 
copper  to  the  zinc,  in  the  direction  that  is  in  which  the  current 
is  flowing,  the  electromotive  force  is  Fi  -  F2,  and  the  resistance 
R±,  hence  if  C  be  the  current 


or  Vl  -  Yt  =  UJKlt 

14—2 


212  ELECTRICITY  [CH.  XIV 

But  as  we  pass  from  the  zinc  to  the  copper  through  the 
battery,  still  going  with  the  current,  the  electromotive  force  is 
F2  +  E  —  Vlt  the  resistance  is  #2,  and  the  current  C. 

Hence  if  we  suppose  the  same  law  to  hold 


or  ^-(ri-F2 

adding  these  two  results  we  find 


if  R  =  Rl  +  R2, 

and  this  may  be  written  in  the  alternative  forms 


=. 
In  the  above  case  TjJis  denned  as  the  resistance  of  the  circuit. 

136.  Ohm's  Law.  When  the  current  is  due  to  the 
action  of  a  battery  we  can  to  a  certain  extent  localize  the  seat 
of  the  B.M.F.  —  it  is  at  any  rate  somewhere  in  the  battery  though 
it  may  be  difficult  to  say  at  exactly  what  point  it  acts;  there 
are  however  many  cases  in  which  the  electromotive  force  acts 
continuously  at  all  points  of  the  circuit,  still  experiment  shews 
that  in  all  cases  the  law  connecting  the  three  quantities  E.M.F., 
current,  and  resistance  is  the  same.  This  law,  which  is  called 
from  its  discoverer  Ohm's  law,  may  be  stated  thus  : 

OHM'S  LAW.  In  any  circuit  the  ratio  of  the  electromotive 
force  producing  a  current  to  the  current  prod^lced  is  a  constant 
depending  only  on  the  form,  materials,  and  physical  conditions 
of  the  circuit.  This  constant  is  called  the  Resistance  of  the 
circuit. 

We  may  express  Ohm's  Law  in  symbols  thus:  —  If  E  be  the 
electromotive  force,  C  the  current,  and  R  the  resistance,  then 

E-R 

C~ 

and  R  is  constant  for  the  circuit. 


135-138]      ELECTROMOTIVE    FORCE   AND   CURRENT          213 

137.  Unit  of  Resistance.     Electrical  resistance,  like 
other  quantities,  is  measured  in  terms  of  a  proper  unit,  and 
this  unit  is  called  an  "  Ohm." 

The  definitions  of  the  Ohm,  the  Ampere,  and  the  Volt, 
are  based  ultimately  on  certain  theoretical  considerations,  and 
are  connected  together  in  such  a  way  that  an  ampere  is  the 
current  produced  by  one  volt  acting  through  a  resistance  of 
one  ohm.  From  this  it  follows  that  if,  in  the  equation 
representing  Ohm's  law,  we  measure  E  and  R  in  volts  and 
ohms^?then  C  is  measured  in  amperes. 

Thus  for  example  if  the  resistance  of  a  circuit  is  50  ohms  and  an 
E.M.F.  of  5  volts  acts  round  it,  the  current  is  5/50  or  1/10  of  an  ampere  ;  or 
again,  if  the  current  is  5  amperes  and  the  resistance  50  ohms,  the  E.M.F. 
is  5  multiplied  by  50  or  250  volts. 

Since  the  resistance  of  a  conductor  is  a  physical  property 
of  the  conductor  depending  on  its  material,  shape  and  con- 
ditions, any  given  conductor,  such  as  a  piece  of  wire,  has  a 
definite  resistance  in  ohms. 

A  column  of  mercury  106 '3  centimetres  in  length,  and 
1  square  millimetre  in  cross  section,  has  been  found  to  have, 
when  at  the  temperature  of  melting  ice,  a  resistance  very 
approximately  equal  to  that  of  the  ohm  as  theoretically 
denned,  and  it  has  been  agreed  to  take  the  resistance  of 
such  a  column  as  the  practical  unit  of  resistance  and  call 
it  one  ohm. 

In  order  to  get  over  certain  difficulties  of  measurement, 
the  column  is  defined  by  its  length,  and  the  mass  of  mercury 
it  contains  at  zero  Centigrade,  instead  of  by  its  length  and 
cross  section.  Accordingly  the  following  practical  definition 
has  been  generally  agreed  upon. 

DEFINITION.  A  column  of  mercury  of  uniform  cross  section, 
106 -3  centimetres  in  length,  which  contains  at  the  temperature 
of  melting  ice  a  mass  of  14*4521  grammes  of  mercury  has  a 
resistance  of  an  Ohm. 

An  electromotive  force  of  one  volt,  applied  to  the  ends 
of  such  a  column,  will  produce  a  current  of  one  ampere. 

138.  Conductance.      If    R    be   the   resistance   of  a 
conductor,  the  quantity  \JR  is  known  as  its  Conductance. 


214  ELECTRICITY  [CH.  XIV 

It  is  the  ratio  of  the  current  in  the  conductor  to  the  electro- 
motive force  producing  it.  The  conductance  of  a  conductor 
whose  resistance  is  1  ohm  is  unity,  that  of  a  conductor  whose 
resistance  is  50  ohms  is  1/50,  and  so  on. 

139.     Conductors  in  series. 

PROPOSITION  13.  To  find  the  resistance  of  a  number  of 
conductors  in  series. 

Let  A-^A^.  A2AS)  ASA4)  etc.,  Fig.  135,  be  a  series  of 
conductors  connected  together  at  A 2,  A3,  etc.  Let  Flf  F2,  F3,  ... 
be  the  potentials  at  A^  A2 ... ,  7^,  7?2,  ...  the  resistances  of  the 
conductors,  and  let  a  current  C  traverse  this  series. 


Then  we  have  Vl  -  F2  =  C 


'n  ~  'n+i  — 


Hence  adding  these 


But  if  R  denotes  the  resistance  between  the  extreme 
points  A^  and  An+l  between  which  the  current  C  is  flowing, 

then  Vl-Vn+l  =  CR. 

Hence  R  =  Rl+...Rn. 

In  other  words  the  resistance  of  a  number  of  conductors 
connected  so  that  the  same  current  traverses  each  in  turn,  is 
the  sum  of  the  resistances  of  the  various  conductors. 

^  It  follows  clearly  from  this  that  the  resistance  of  a  piece 
of  wire  of  uniform  material  and  thickness,  is  proportional  to 
the  length  of  the  wire.  For  let  the  wire  be  I  cm.  in  length, 


138-140]      ELECTROMOTIVE    FORCE   AND   CURRENT          215 

we  may  consider  it  as  I  equal  conductors  in  series,  each  of 
them  being  1  cm.  in  length.  Let  a-  be  the  resistance  of  1  cm. 
of  the  wire,  then  the  resistance  of  the  I  conductors  in  series, 
each  of  a  resistance  <r,  is  Icr.  Thus  the  whole  resistance  is  la- ; 
it  is  therefore  proportional  to  the  length  of  the  wire. 

14O.  Conductors  in  parallel.  When  two  or  more 
conductors  join  the  same  two  points  A,  A',  so  that  a  current 
from  A  to  A'  can  flow  by  two  or  more  paths,  the  conductors 
are  said  to  be  in  parallel,  or  sometimes  in  multiple  arc. 

PROPOSITION  14.  To  find  the  resistance  of  a  number  of 
conductors  in  parallel. 

Let  AA^A,  AA»A',  etc.,  Fig.  136,  be  a  number  of  con- 
ductors joining  two  points  A  and  A'.  Let  a  current  C  be  led 


Fig.  136. 

into  the  system  at  A  and  withdrawn  at  A',  and  let  Cl,  C2,  Cs 
be  the  currents  in  the  various  conductors,  R1,  R%,  etc.  the 
resistances  of  the  conductors.  Let  V,  V  be  the  potentials 
at  A  and  A.  Now  the  sum  of  the  currents  in  the  various 
paths  between  A  and  A'  is  equal  to  the  current  led  in  at  A. 

Hence  C  =  Cl  +  C2  +  . . .  +  Cn . 

V-V 

Also  C1  =- 


C2  = 

etc. 
Hence 


216  ELECTRICITY  [CH.  XIV 

And  if  R  is  the  equivalent  resistance,  the  resistance  that 
is  of  a  conductor  which  under  the  potential  difference  V  —  V 
will  permit  of  the  passage  of  the  current  (7, 

then  C-&=. 


111  1 

Hence  -  =  -  +  -  +  ...+  «-, 

R     HI      M2  Kn 

or  in  words: 

The  conductance  of  a  number  of  conductors  in  parallel  is 
the  sum  of  the  conductances  of  the  several  conductors. 

•vj  It  follows  from  this  that  the  conductance  of  a  wire  is 
proportional  to  the  area  of  its  cross  section.  For  consider  a 
wire  of  any  length  I  cm.  and  a  sq.  cm.  in  area  of  cross  section. 
We  may  split  it  up  into  a  wires  each  1  sq.  cm.  in  area  placed 
side  by  side.  Let  the  resistance  of  each  centimetre  of  each 
of  these  wires  be  p,  then  the  resistance  of  each  wire  is  pi, 
and  the  conductance  of  each  wire  is  I/  pi',  we  have  now  a 
wires  in  parallel,  each  of  conductance  1  /pi.  It  follows  that 
the  conductance  of  the  whole  is  aj  pi  ;  that  is,  the  conductance 
is  proportional  to  the  area  of  the  cross  section. 

If  R  be  the  resistance  of  the  wire,  then  we  have  seen  that 
1  _  a 
R~  pi' 

Hence  R  =  *-  . 

a 

141.  Specific  Resistance1.  Thus  the  resistance  of  a 
uniform  wire  is  proportional  to  its  length  and  inversely 
proportional  to  the  area  of  its  cross  section.  The  quantity  p 
is  the  resistance  of  a  piece  of  the  wire  1  cm.  in  length,  having 
an  area  of  1  sq.  cm.  in  cross  section  :  the  form  of  this  cross 
section  is  not  material  ;  if  we  imagine  it  to  be  square  we  see 
that  p  is  the  resistance  between  two  opposite  faces  of  a  cube 
each  edge  of  which  is  1  centimetre.  This  quantity  is  called 
the  Specific  Resistance  of  the  material  of  the  wire. 

1  It  should  be  noticed  that  the  term  specific  is  used  in  its  proper 
sense  and  does  not  imply  a  ratio  as  in  "specific"  heat,  "specific" 
gravity. 


140-142]      ELECTROMOTIVE    FORCE   AND   CURRENT  217 

DEFINITION.  The  Specific  Resistance  of  a  material  is 
the  resistance  between  two  opposite  faces  of  a  cube  of  the  material 
each  edge  of  which  is  1  centimetre  in  length. 

Thus  we  have  the  result  that  if  p  be  the  specific  resistance 
of  a  wire  of  length  I  cm.  and  cross  section  a  sq.  cm.,  and  R  the 
resistance  of  the  whole  wire, 

then  R.=  ~  ohms. 

V    a 

Conversely  if  we  are  given  the  resistance,  length  and  cross 
section  of  a  wire  we  can  find  the  specific  resistance  of  its 
material  from  the  formula 

Ra. 


The  following  Table  gives  the  specific  resistance  of  a 
number  of  materials  in  microhms1  per  cube  centimetre  and  the 
resistance  in  ohms  of  a  length  of  1  metre,  1  square  millimetre 
in  cross  section  at  0°  C. 

TABLE. 

Kesistance  of  1  metre 

Specific  Resistance  1  sq.  mm.  in  area  of 

Material  in  Microhms  cross  section  in  Ohm 

.    Silver  annealed  1-468  -01468 

„      hard  drawn  1-615  -01615 

Gold  annealed  2-036  -02036 

Zinc  5-751  -05751 

Copper  annealed  1-562  -01562 

hard  drawn  1-603  -01603 

Iron  9-065  -09065 

Platinum  10-917  -10917 

Mercury  94-073  -94073 

Platinum  Silver  24-120  -24120 

German  Silver  20-243  -20243 

Manganin  46-700  -46700 

Eesista  76-490  -76490 

142.  Distribution  of  current  between  a  number 
of  conductors  in  parallel.  The  question  may  often  arise 
as  to  how  a  current  will  distribute  itself  in  a  series  of 
conductors  in  parallel.  The  principles  of  the  foregoing  pro- 
position enable  us  to  answer  this. 

1  A  microhm  is  a  millionth  of  an  ohm  or  10~6  ohms. 


218  ELECTRICITY  [CH.  XIV 

PROPOSITION  15.  To  find  the  distribution  of  current  in  a 
number  of  conductors  in  parallel. 

As  in  Fig.  136  above,  let  AA^A',  AA2A',  etc.  be  the 
conductors.  Let  C  be  the  current  entering  at  A  and  leaving 
at  A'  ;  while  (7a,  <72,  <?3  ...  are  the  currents,  R^  JR2,  Rz  ...  the  re- 
sistances of  the  conductors.  Then  since  the  potential  difference 
between  A  and  A'  is  the  same  for  each  of  the  possible  paths, 
and  since  the  potential  difference  is  measured  by  the  product 
of  the  current  and  the  resistance  we  must  have 


and  .... 

- 
Hence 

Thus 


etc. 

where  \/R  is  the  conductance  of  the  system,  being  given  by 
the  equation 

1  -1      1  A 

R     Rl     R%          Rn 

If  there  be  two  circuits  only  of  resistances  R^  and  R2 
1=1       1 

it  Jit}  41,) 


c= 


This  result  is  important  in  the  theory  of  galvanometers,  §  159. 

143.     Graphic    representation    of   Ohm's    Law. 

Let    us   suppose   that  on  a  diagram   we  represent  potential 


142-144]       ELECTROMOTIVE    FORCE   AND   CURRENT          219 

differences  by  vertical  straight  lines,  and  resistances  by 
horizontal  straight  lines.  Thus  iu  Fig.  137,  let  PM  re- 
present the  electromotive  force  E  round  a  given  circuit,  and 
let  MN  represent  R  the  resistance  of  the  circuit. 


M 


Fig.  137. 

Then  the  current  C  which  is  given  by  the  ratio  E/R  is 
represented  by  PMJMN  or  by  tan  PNM. 

144.     Chemical  Theory  of  a  Voltaic  Cell,    If  the 

circuit  consist  of  a  series  of  different  materials,  and  we  know 
how  the  potential  changes  as  we  pass  from  one  material  to 
the  next  we  can  draw  a  similar  diagram.  Thus  in  the  case 
of  a  voltaic  cell,  there  is  in  the  views  of  many  electricians 
reason  to  suppose  that  when  the  zinc  is  dipped  into  the 
dilute  acid,  its  potential  is  less  than  that  of  the  acid  by 
1*8  volts,  while  the  potential  of  the  copper  falls  below  that 
of  the  acid  by  about  -8  of  a  volt.  If  a  piece  of  copper  wire 
be  connected  to  the  zinc  it  will  be  at  the  same  potential  as 
the  zinc.  Thus  when  the  circuit  is  open  the  distribution  of 
potential  is  as  shewn  in  Fig.  138. 


Fig.  138. 

If  the  circuit  be  completed  by  joining  the  free  end  of  tho 
copper  wire  to  the  copper  plate  we  proceed  thus  : 


220 


ELECTRICITY 


[CH.  XIV 


Let  OA,  Fig.  139,  represent  the  resistance  of  the  zinc,  AB  of 
the  acid,  BC  of  the  copper,  CD  of  the  wire.  Draw  ON  perpen- 
dicular to  OABCD  to  represent.^  the  E.M.F.  of  the  cell  and 
join  ND.  Then  the  current  is  represented  by  the  tangent 
of  the  angle  NDO. 


Fig.  139. 

Let  PAQ  and  BSR  be  drawn  parallel  to  ON.  Draw  OP 
parallel  to  ND  to  meet  PAQ  in  P,  and  take  PQ  to  represent 
the  difference  of  potential  between  the  zinc  and  the  acid. 
From  Q  draw  QR  parallel  to  ND  to  meet  BSR  in  R. 

Let  PQ  meet  ND  in  T. 

Then  RS=QT. 

Also  PT  =  ON=  E.M.F.  of  cell. 

But  PQ  is  the  potential  difference  between  the  zinc  and 
the  acid,  hence  QT  is  the  potential  difference  between  the 
acid  and  the  copper.  Thus  if  the  point  R  represents  the 
potential  of  the  acid  in  contact  with  the  copper,  S  represents 
the  potential  of  the  copper. 

Hence  if  the  point  0  represents  the  potential  of  the 
point  of  contact  of  the  copper  and  zinc,  the  curve  OPQRSD 
represents  the  distribution  of  potential. 

The  current  which  is  uniform  throughout  is  represented 
by  the  tangent  of  the  inclination  of  the  parts  OP,  QR,  SD 
to  the  resistance  line ;  the  potential  falls  uniformly  along  OP, 
through  the  zinc,  then  there  is  a  sudden  rise  at  the  zinc 


144-145]      ELECTROMOTIVE   FORCE   AND   CURRENT          221 

surface  to  PQ,  a  uniform  fall  from  AQ  to  BR  through  the 
acid,  a  sudden  drop  R8  at  the  copper  surface,  and  a  uniform 
fall  along  SD  from  SB  to  the  original  value,  through  the 
copper  and  copper  wire. 

Moreover  let  CU  parallel  to  ON  meet  SD  in  U  ;  if  the 
ends  of  the  external  circuit  represented  by  C  and  D  be 
connected  to  an  electrometer,  the  potential  difference  registered 
will  be  represented  by  CU.  When  the  battery  was  on  open 
circuit  the  potential  difference  was  ON  the  E.M.F.  of  the 
battery,  and  we  see  hence  how  it  is  that  the  difference  of 
potential  between  the  plates  of  a  battery  falls,  as  stated 
in  Section  107,  when  the  circuit  is  closed  through  an  external 
resistance. 

If  we  call  E  the  E.M.F.  of  the  battery,  E^  the  potential 
difference  between  the  ends  of  the  external  resistance,  R  the 
whole  resistance,  and  Rl  the  external  resistance,  then  in  the 
figure 

ON=E,  CU=E^ 


and  OC  or  R  —  Rl  is  the  battery  resistance. 
Then  we  have  clearly 

Current  =  tan  NDO 


145.  Contact  theory  of  a  Voltaic  cell.  According 
to  another  theory  of  the  cell,  the  copper,  zinc,  and  acid, 
when  on  open  circuit  are  all  at  very  nearly  the  same  potential, 
but  if  a  copper  wire  be  connected  to  the  zinc,  a  difference 
of  potential  is  established  between  the  zinc  and  the  copper, 
the  potential  of  the  zinc  exceeding  that  of  the  copper  by 
about  1  volt,  and  this  measures  the  electromotive  force  of 
the  cell. 

In  this  case  the  open  circuit  distribution  is  as  in  Fig.  140. 
When  the  free  end  of  the  copper  wire  is  connected  to  the 
copper  plate  its  potential  is  raised  to  that  of  the  plate  and 


222 


ELECTRICITY 


[CH.  xiv 


a  current  runs  through  the  wire  from  the  copper  to  the  zinc ; 
the  distribution  of  potential  is  as  in  Fig.  141,  where  0  represents 
the  potential  of  the  copper  wire  at  its  junction  with  the  zinc, 


Fig.  140. 

N  is  the  potential  of  the  zinc  where  it  is  joined  by  the  wire, 
and  the  fall  of  potential  through  the  cell  and  wire  is 
by  the  straight  line  NQRUD. 


Fig.  141. 

In  this  case  also  we  clearly  have,  as  before, 

E     El     E-E, 
Current  - 


The  theory  of  the  cell  and  the  experiments  on  which  it 
is  based  are  given  in  Sections  191  —  196. 


CHAPTEE  XV. 


MEASUREMENT   OF  CURRENT. 

146.  Measurement  of  a  Current.  Galvano- 
meters. A  current  may  be  measured  either  by  its  magnetic, 
its  chemical,  or  its  thermal  effects.  Various  instruments  have 
been  devised  for  utilizing  the  magnetic  force  due  to  a  current 
to  measure  it.  These  are  called  galvanometers. 

We  have  seen  that  there  is  magnetic  force  exerted  in  the 
neighbourhood  of  a  wire  carrying  a  current ;  if  we  bring  such 
a  wire  near  a  compass-needle  it  is  in  general  deflected ;  the 
deflexion  is  increased  by  bringing  the  needle  near  to  the  wire ; 
it  is  also  increased  by  strengthening  the  current.  Bend  the 
wire  into  the  form  of  a  circle  some  8  or  10  cms.  in  radius, 
and  hold  it  in  a  vertical  plane  so  that  the  point  of  support 
of  the  compass-needle  is  at  its  centre  and  the  plane  of  the 
coil  is  north  and  south,  the  compass-needle  is  deflected  by 
the  current.  If  the  number  of  turns  of  wire  in  the  coil  be 
increased  or  its  radius  decreased,  the  deflexion  of  the  needle 
is  increased.  In  all  cases  there  is  magnetic  force  exerted  by 
the  current  and  it  remains  to  measure  the  force. 

Now  it  is  found  that  if  a  length  I  of  wire  be  bent  into 
the  form  of  an  arc  of  a  circle  of  radius  r,  and  if  a  current  i 
is  allowed  to  flow  through  the  wire,  the  magnetic  force  at 
the  centre  of  the  arc  is  perpendicular  to  the  plane  of  the 
arc  and  is  proportional  to  il/r2.  The  arc  may  be  less  than  a 
complete  circle  or  it  may  include  one  or  more  complete  turns ; 
if  it  include  one  turn  exactly,  then  I  =  2-n-r,  and  the  force  is 
proportional  to  ^Trri/r"  or  2-jri/r.  If  it  include  n  turns  then 


224 


ELECTRICITY 


[CH.  XV 


I  -  2mrr  and  the  force  is  then  proportional  to  2mri/r.  In 
either  of  these  cases  lines  of  magnetic  force  are  produced 
which  stream  through  the  coil  and  are  linked  with  it. 

Let  the  coil  be  placed  in  a  vertical  plane,  and  fix  a 
horizontal  sheet  of  card  or  glass  so  as  to  pass  through  its 
centre.  Sprinkle  the  sheet  with  iron  filings;  on  passing  a 
current  through  the  coil  and  tapping  the  sheet,  the  iron 
filings  set  along  the  .lines  of  force  as  shewn  in  Fig.  142, 


Fig.  142. 

in  which  the  dotted  lines  are  lines  of  force,  the  strong  lines 
equipotential  surfaces.  The  lines  could  be  mapped  out  on 
the  sheet  by  means  of  a  little  compass-needle  as  described 
in  Section  71. 

In  doing  this  it  must  be  remembered  that  the  magnet  is 
also  acted  on  by  the  earth's  magnetism. 

147.     Unit  of  Current.     If  we  assume  the  above  law 
to  hold  we  can  deduce  certain  important  consequences  from 


146-148]  MEASUREMENT  OF   CURRENT 

it.  The  law  stated  that  the  force  was  proportional  to  tins 
quantity.  Let  us  assume  the  proportion  to  be  one  of  equality, 
and  let  us  farther  suppose  that  the  current  is  measured  in 
such  a  way  that  we  may  write 

U 
F—  Magnetic  force  at  centre  =  -3. 

v  let  the  length  of  the  wire  be  1  cm.  and  the  radius  of 
the  circle  1  cm.,  then 

1=1,  r  =  \. 

Hence  F- 

Thus  the  measure  of  the  current  is  in  this  case  equal  to 
that  of  the  magnetic  force  which  it  exerts  at  the  centre  of  the 
coil.  If  this  force  be  the  unite  force  so  that  a  magnetic  pole  of 
unit  strength  placed  at  the  centre  of  the  circle  is  acted  on  by 
a  force  of  one  Dyne,  then  we  have  F=l,  and  therefore  *  =  1, 
or  the  current  is  the  unit  of  current.  This  unit  is  known  as 
trojnagnetic  Unit  of  Current. 

DEFINITION.  The  Electromagnetic  Unit  of  Current  is  a 
current  which,  flowing  in  a,  wire  one  centimetre  in  length,  bent 
into  the  form  of  an  arc  of  a  circle  one  centimetre  in  radius,  pro- 
duces unit  magnetic  force  at  the  centre  of  the  circle. 

The  quantity  of  electricity  which  is  conveyed  by  this 
current  in  unit  time  across  each  section  of  the  conductor  in 
which  it  is  flowing  is  the  electromagnetic  unit  quantity  of 
electricity. 

This  electromagnetic  unit  is  found  by  experiment  to  be 
very  much  greater  than  the  electrostatic  unit.  Each  electro- 
magnetic unit  contains  3  x  1<P  electrostatic  units. 

It  should  be  noted  that  the  definition  of  the  electrostatic  unit  is  baaed 
on  the  supposition  that  the  inductive  capacity  of  air  is  unity,  thai  of  the 
electromagnetic  unit  depending  as  it  does  on  the  unit  of  magnetic  force 
assumes  the  magnetic  permeability  of  air  to  be  unity,  §  75. 

148.  Practical  Unit  of  Current.  The  unit  of  current 
selected  for  practical  purposes  is,  as  we  have  said,  the  ampere. 
The  ampere  is  defined  in  terms  of  the  number  of  grammes  of 
silver  which  are  deposited  by  it,  per  second,  from  a  solution  of 
nitrate  of  silver.  It  has  been  found  as  the  result  of  various 

G.    E.  15 


226 


ELECTRICITY 


[CH.  XV 


experiments  that  the  passage  of  the  electromagnetic  unit  of 
current  for  1  second  causes  the  deposit  of  -01118  gramme  of 
silver.  One-tenth  of  this  current  is  taken  as  a  practical  unit, 
and  is  called  an  ampere. 

J  DEFINITION.  A  current  which  deposits  per  second  -001118 
gramme  of  silver  from  a  neutral  solution  of  nitrate  of  silver  in 
water  is  called  one  Ampere. 

The  quantity  of  electricity  conveyed  by  an  ampere  flowing 
for  one  second  is,  as  we  have  already  said,  §  109,  called  a 
Coulomb.  A  Coulomb  is  one-tenth  of  the  electromagnetic 
unit  quantity  of  electricity. 

149.  Galvanometers.     If  a  current  traverse  a  wire 
bent  into  the  form  of  a  circle  the  direction  of  the  magnetic 
force  exerted  near  the  centre  of  the  circle  is  at  right  angles  to 
its  plane.     If  r  cm.  be  the  radius  of  the  circle  the  force  at 
the  centre  due  to  a  current  i  is  2iri/r.     The  force  at  a  point 
near  the  centre  is  (if  the  circle  is  not  too  small)  given  approxi- 
mately by  this  expression,  and  if  a  small  magnet  with  poles 
of  strength  m  be  suspended  at  the  centre  each  pole  is  acted  on 
by  a  force   2nnri/r.     These  forces  act  in  opposite  directions 
perpendicular  to   the  plane  of   the   coil. 

Each  pole  of  the  magnet  will  be  acted 
on  also  by  a  force  mil,  where  H  is  the 
strength  of  the  earth's  field,  and  the 
magnet  will  take  up  a  position  depending 
on  the  relation  between  these  forces. 

150.  Tangent  Galvanometer. 

Let  us  now  suppose  that  the  plane  of 
the  coil  is  north  and  south,  and  that  the 
current  above  the  magnet  is  running 
from  south  to  north,  below  it  from  north 
to  south.  The  force  on  the  north  pole  of 
the  magnet  will  be — applying  the  right- 
handed  screw  rule — from  east  to  west 
and  the  north  pole  therefore  will  be 
deflected  toward  the  west.  Let  F  (Fig. 

143)  be  the  magnetic  force  toward  the  west,  then  the  magnet 
is  in  equilibrium  under  two  couples  of  which  the  forces  are  at 


Fig.  143. 


148-150]  MEASUREMENT   OF   CURRENT  227   « 

right  angles,  the  one  F  toward  the  west  and  the  other  H 
toward  the  north  (§  92).  The  tangent  law  holds,  and  if  0 
be  the  angle  the  magnet  is  deflected  from  the  magnetic 
meridian  we  have 

mFcos  0  —  mffsin  0 


or 

Now  we  have  already  seen  that  if  the  current  traverses 
a  single  turn  of  a  circle 


r 

Hence  —  =H  tan  9. 

r 

Ifr 
Therefore  i  =  -^-  tan  0. 

If  there  are  n  turns  in  the  coil  traversed  by  the  current 


r 

and  i  =  -=  —  tan  0. 

2mr 

An  instrument  of  this  kind  is  called  a  tangent  galvano- 
meter. 

Thus  if  we  know  the  value  of  //  we  can  by  measuring  r, 
the  radius  of  the  circle  in  which  the  current  flows,  and 
observing  the  value  of  $,  the  deflexion  of  the  magnet,  calculate 
i  the  strength  of  the  current,  from  the  above  formula. 

The  current  so  obtained  will  be  in  electromagnetic  units. 
Since  one  ampere  is  one-tenth  of  such  a  unit,  in  order  to  fifid 
the  current  in  amperes  we  must  multiply  by  10. 

We  thus  have 

IQ.ffr 

i=     —    -    tan  0,    amperes. 

2mr 

Example.  Having  given  that  the  value  of  H  is  '18  unit  and  that  the 
radius  of  a  coil  of  IQ  turns  is  5  cms.,  find  the  current  in  amperes  which 
will  cause  a  deflexion  of  45°. 

In  this  case  since  tan  45  =  1,  we  have 
10  x  -18  x  5 

'=^rroxV=-143ampere- 

15—2 


228  ELECTRICITY  [CH.  XV 

151.  Reduction  Factor.    Sometimes  it  is  convenient 
to   denote   the   quantity   Hrfinv,    or   if   we  are   working  in 
amperes  the  quantity  lOHr/^mr,  by  a  single  symbol  k. 

We  then  have  i  =  k  tan  0. 

The  quantity  k  is  known  as  the  reduction  factor  of  the 
galvanometer ;  it  depends  on  its  construction  and  on  the  value 
of  H  at  the  spot  where  it  is  used  but  not  on  the  deflexion. 

In  the  case  of  a  tangent  galvanometer  it  may  be  denned  as 
the  quantity  by  which  the  tangent  of  the  deflexion  must  be 
multiplied  in  order  to  give  the  current. 

If  we  know  the  current  required  to  give  a  certain  deflexion 
we  can  find  the  reduction  factor  by  dividing  the  current  by 
the  tangent  of  the  deflexion. 

Again,  since  i  =  k  tan  0  and  tan  45°  is  unity  we  see  that  if 
0  =  45°  we  have  i  =  k.  Thus  the  reduction  factor  is  measured 
by  the  current  required  to  produce  a  deflexion  of  45°. 

Example.  A  current  of  10  amperes  produces  a  deflexion  of  60°,  find 
the  current  tvhich  will  produce  a  deflexion  0/30°. 

Since  i  =  k  tan  0 

we  have  10=  A;  tan  60. 

Hence  fc  =  10/tan  60  =  10/^3  =  5 '78, 

and  the  current  required  =  5 -78  tan  30  =  5*78/^3  — 3*33  amperes. 

\  Or  more  simply,  since  the  currents  are  proportional  to  the  tangents  of 

the  deflexions,  if  i  be  the  required  current 

i_  _  tan  30  _  1 
10  ~  tan  60  ~  3  ' 

Hence  i  =  _  =  3  -33  amperes. 

o 

152.  Galvanometer  Constant.     We  have  seen  that 
the  force  exerted  at  the  centre  of  the  coil  by  a  current  i  is 
2niri/rt   so  that  the  force  exerted  by  unit  current  is  2nir/r. 
This  quantity  is  called  the  galvanometer  constant  of  the  coil. 
It  is  often  denoted  by  the  symbol  G,  so  that  for  a  circular  coil 
of  n  turns  we  have 


151-153]  MEASUREMENT   OF   CURRENT 


229 


If  the  coil  be  not  circular  but  have  some  other  symmetrical 
shape,  the  magnetic  force  at  its  centre  due  to  a  unit  current  is 
still  called  the  galvanometer  constant,  though  its  value  cannot 
be  so  simply  expressed  as  in  the  case  of  a  circle.  In  any  case 
the  force  at  the  centre  due  to  a  current  i  is  Gi  in  the  direction 
of  the  axis,  and  if  0  be  the  deflexion  of  the  magnet  we  have 

Gi=Htan6 

TT 

or  i  =  7^  tan  6. 

\JT 

Comparing  this  with  the  equation 

i  =  k tan  6 
we  see  that  k  =  ff/G. 

Thus  the  reduction  factor  is  measured  by  the  ratio  of  the 
strength  of  the  earth's  field  to  the  galvanometer  constant. 

153.  Sine  Galvanometer.  In  the  tangent  galvano- 
meter the  coil  is  placed  in  the  mag- 
netic meridian  so  that  it  is  parallel 
to  the  needle  when  there  is  no  current 
in  the  circuit.  The  sine  galvanometer 
is  arranged  so  that  the  coil  can  be 
turned  round  a  vertical  axis  through 
its  centre.  When  a  current  traverses 
the  coil  the  needle  is  deflected  and 
the  coil  is  turned  so  as  to  follow  the 
needle  until  its  plane  is  again  parallel 
to  the  needle  in  its  deflected  position. 
Thus  the  magnetic  force  due  to  the 
current  in  the  coil  acts  at  right  angles 
to  the  magnet.  If  we  call  this  force 
Gi,  and  if  6  be  the  deflexion  as  before 
(Fig.  144),  by  taking  moments  round  the  centre  of  the  magnet 
we  obtain 

Gi  =  H  sin  0. 


Fig.  144. 


Therefore 
or  writing 


•  -     /> 

i  =       sin  0, 


i  =  k  sin  6. 


230 


ELECTRICITY 


[CH.  XV 


For  a  circular  coil  the  value  of  G  is  Smr/r.  Thus  the 
current  is  in  this  case  measured  by  the  sine  of  the  deflexion 
and  the  instrument  is  a  sine  galvanometer. 

It  must  be  noted  that  in  both  these  instruments  it  is  assumed  that 
the  force  acting  on  the  magnet  is  determined  by  its  value  at  the  centre 
of  the  coil.  This  at  best  is  only  approximately  true,  and  it  is  necessary 
that  the  magnet  should  be  small  compared  with  the  radius  of  the  coil. 

Again,  in  the  formula  F=2irijr,  r  is  the  radius  of  a  single  turn  in 
which  the  current  flows,  and  the  magnet  hangs  at  its  centre  ;  when  the 
coil  has  a  number  of  turns,  these  are  usually  arranged  in  layers ;  the 
magnet  clearly  cannot  be  at  the  centre  of  all  the  coils  in  a  given  layer 
and  the  radii  of  the  coils  in  different  layers  are  different.  Hence  in 
the  formula  F=2mri/r,  r  has  to  be  taken  as  the  mean  radius  of  the  coil ; 
if  however  the  dimensions  of  a  section  of  the  coil  are  small  compared 
with  its  radius,  the  mean  radius  can  be  determined  with  considerable 
accuracy. 

154.    Construction  of  a  Tangent  Galvanometer. 

A  tangent  or  sine  galvanometer  then  usually  consists  of  a 
circular  coil  of  insulated  wire  wound 
in  layers  in  a  suitable  groove,  as 
shewn  in  Fig.  145.  The  coil  is 
carried  on  levelling  screws  and 
mounted  so  that  it  can  be  rotated 
round  a  vertical  axis  ;  a  graduated 
circle  is  attached  to  the  coil,  the 
planes  of  the  two  being  at  right 
angles,  so  that  when  the  coil  is 
vertical  the  circle  is  horizontal ;  the 
centres  of  the  two  coincide,  and 
at  the  centre  of  the  circle  is  fixed 
a  pivot  which  carries  a  small  mag- 
netic needle.  A  light  pointer  is 
attached  to  this  and  moves,  as  the 
needle  swings,  over  the  graduated 
circle ;  this  pointer  is  usually  fixed 
at  right  angles  to  the  axis  of  the  magnet. 

The  circle  is  usually  graduated  so  that  when  the  axis 
of  the  magnet  is  in  the  plane  of  the  coil  the  pointer  reads 
zero  on  the  circle.  In  using  the  instrument  both  ends  of 
the  pointer  are  read.  This  eliminates  the  error  which  might 
otherwise  arise  from  a  want  of  exact  centering  .of  the  circle. 


Fig.  145. 


153-154]  MEASUREMENT   OF   CURRENT  231 

The  magnet  and  pointer  are  protected  from  air  currents  by 
a  transparent  cover. 

In  a  sine  galvanometer  a  second  horizontal  circle  is  usually 
attached  to  the  base  of  the  instrument,  and  on  this  the  angle 
through  which  the  coil  is  turned  can  be  read. 

The  ends  of  the  wire  are  secured  to  binding  screws  on  the 
base  and  the  current  is  led  into  these.  Sometimes  a  number 
of  different  circuits  are  wound  in  the  same  groove ;  the  range 
of  the  instrument  is  thereby  increased ;  for  large  currents  a 
single  turn  may  suffice  to  produce  a  measurable  deflexion, 
for  weaker  currents  a  large  number  of  turns  may  be  necessary. 
Moreover  a  single  turn  of  thick  wire  will  have  a  low  resistance 
which  may  be  desirable  for  some  purposes,  while  for  others 
a  coil  of  thin  wire  having  a  large  number  of  turns  and  in 
consequence  a  considerable  resistance  may  be  required. 

To  use  the  instrument  as  a  tangent  galvanometer  it  is 
levelled  so  that  the  coil  is  vertical  and  the  needle  swings  freely  ; 
then  the  coil  is  turned  until  it  is  parallel  to  the  magnet, 
and  in  this  case  if  the  adjustments  are  correct  both  ends  of 
the  pointer  read  zero,  or  if  the  graduations  run  continuously 
round  the  circle,  one  end  reads  zero  and  the  other  180°. 
In  this  position  the  force  exerted  by  the  current  will  be 
perpendicular  to  that  due  to  the  earth.  The  current  is  then 
allowed  to  traverse  the  coil,  thus  deflecting  the  needle;  as 
it  comes  to  rest  the  instrument  is  gently  tapped  to  reduce 
the  effects  of  friction.  Both  ends  of  the  needle  are  again 
read  and  the  deflexion  Ol  is  obtained  by  taking  the  mean 
of  the  deflexions  given  by  the  two  ends ;  then  if  the  reduction 
factor  is  known  the  current  i  is  given  by  the  formula 

i  =•  k  tan  9l . 

If  the  current  can  be  reversed  through  the  coils,  this  is  done 
and  another  deflexion  02  is  found;  the  values  of  Ol  and  #2, 
if  the  adjustments  are  perfect,  will  be  the  same,  but  the  small 
errors  which  may  arise  owing  to  (1)  the  pointer  not  being 
accurately  perpendicular  to  the  axis  of  the  magnet,  and 
(2)  the  zero  of  the  scale  being  not  quite  correctly  placed 
are  eliminated  by  taking  their  mean. 

To  use  the  instrument  as  a  sine  galvanometer  it  is  adjusted 
as  before.  * 


I 

232  ELECTRICITY  [CH.  XV 

On  allowing  the  current  to  traverse  the  coil  the  magnet 
is  deflected,  and  the  coil  is  turned  round  until  the  pointer 
again  reads  zero  ;  in  this  position  the  magnet  is  again  parallel 
to  the  coil  and  the  force  produced  by  the  current  is  at  right 
angles  to  the  magnet. 

The  angle  through  which  the  coil  has  been  turned  is  then 
read  on  the  graduated  circle  at  the  base  of  the  instrument. 
This  gives  one  deflexion  9l.  By  re  versing  the  current  and 
turning  the  coil  in  the  opposite  way  we  get  a  second  reading 
02«  Then  the  current  is  given  by 

i  =  k  sin  \  (0j  +  Oa). 

If  there  be  no  circle  on  the  base  of  the  instrument  we  may  proceed 
as  above,  but  after  turning  the  coil  carrying  the  current  until  the  pointer 
reads  zero  the  circuit  is  disconnected,  when  the  magnet  swings  back 
into  the  meridian.  The  mean  of  the  readings  of  the  two  ends  of  the 
pointer  taken  in  this  position  will  clearly  give  the  angle  the  coil  has 
been  turned  through. 

In  order  to  make  the  tangent  galvanometer  more  sensitive  a  mirror 
is  sometimes  attached  to  the  magnet ;  in  this  case  the  mirror  and 
magnet  are  suspended  by  means  of  a  fine  silk  fibre,  or  in  some  cases 
the  fibre  is  of  quartz.  A  lamp  is  arranged  as  described  in  Section  96  so 
that  the  light  reflected  from  the  mirror  may  fall  on  a  scale,  and  the 
deflexion  of  the  magnet  measured  by  observing  the  position  of  the  spot 
on  the  scale. 

In  some  instruments  the  coil  of  wire  is  separated  into  two 
parts  which  are  wound  in  two  separate  grooves.  These  are 
placed  with  their  axes  coincident,  and  the  magnet  hangs 
with  its  centre  on  the  axis,  midway  between  the  two  coils. 
If  the  distance  between  the  coils  be  adjusted  so  as  to  be 
equal  to  the  radius  of  either  coil  the  magnetic  field  produced 
by  the  current  in  the  neighbourhood  of  the  magnet  can  be 
shewn  to  be  far  more  uniform  than  when  a  single  coil  with 
the  magnet  at  its  centre  is  used.  The  theoretical  conditions 
assumed  for  this  instrument  are  more  exactly  fulfilled. 

155.  Sensitive  Galvanometers.  In  a  galvanometer 
the  current  is  measured  by  the  deflexion  of  a  magnet  which  it 
produces. 

An  instrument  in  which  a  large  deflexion  is  produced  by 
a  given  current  is  more  sensitive  than  one  in  which  the 
deflexion  is  small. 


154-155]  MEASUREMENT   OF    CURRENT  233 

Now  we  can  increase  the  deflexion  either  by  increasing 
the  force  which  the  current  can  exert  or  by  decreasing  the 
controlling  force  which  maintains  the  magnet  in  its  undisturbed 
position. 

To  increase  the  force  due  to  the  current  we  bring  the  coils 
of  wire  close  to  the  magnet  and  increase  the  number  of  turns 
in  the  coil.  As  an  increase  in  the  number  of  turns  means, 
for  a  given  wire,  an  increase  in  the  resistance,  it  does  not 
follow  that  a  large  number  of  turns  is  always  an  advantage ; 
the  increase  in  resistance  may,  through  reducing  the  current, 
reduce  the  force  more  than  the  increase  in  the  number  of  turns 
increases  it. 

To  reduce  the  control  we  may  either  (1)  reduce  the  strength 
of  the  field  in  which  the  magnet  hangs,  or  (2)  adopt  an  astatic 
system.  These  methods  may  of  course  be  combined. 

(1)  In  the  description  of  a  tangent  galvanometer  given 
above,    it  has   been    assumed   that  the  needle  hangs  in  the 
magnetic  field  due  to  the  Earth  alone  when  the  current  is  not 
on.     This  is  not  necessary ;   we  can  bring  a  bar  magnet  near 
to  the  instrument  in  such  a  position  as  to  counteract  much 
of  the  force  due  to  earth ;  in   this   way  the  strength  of  the 
field  in  which  the  needle  hangs  can  be  much  reduced  and  the 
deflexion  due  to  a  given  current  proportionately  increased. 

(2)  In  the  astatic  galvanometer  two  magnets  are  employed. 
These  are  rigidly  attached~wilh  their  axes  parallel  and  their 
poles  in  opposite  directions. 

If  the  two  magnets  are  exactly  equal  and  their  axes 
exactly  parallel  it  is  clear  that  when  they  are  suspended  in 
a  uniform  field,  the  couple  on  the  one  magnet  will  exactly 
balance  that  on  the  other,  and  the  resultant  will  be  zero. 
In  practice  this  condition  is  never  secured  exactly,  but  by  an 
arrangement  of  this  kind  the  effect  of  the  controlling  force  is 
greatly  reduced. 

Let  the  two  magnets  be  suspended  one  above  the  other 
and  suppose  that,  as  shewn  in  Fig.  146,  the  galvanometer 
coils  are  placed  so  as  to  surround  the  lower  magnet  only, 
leaving  the  upper  magnet  above  the  coil. 

The   current   in  the  coil  will  tend  to  turn  both  magnets 


234  ELECTRICITY  [CH.  XV 

in  the  same  direction  though  its  effect  on  the  upper  magnet 
will  be  small  compared  with  that  on  the  lower. 

Thus  by  this  means  the  controlling  force  is  considerably 
reduced  and  the  sensitiveness  increased. 


Fig.  146. 


Fig.  147. 


The  accuracy  with  which  the  position  of  the  galvanometer 
needle  can  be  read  is  much  increased  by  the  use  of  a  mirror  as 
already  described  by  which  a  beam  of  light  is  reflected  on 
to  a  scale.  Figs.  147  and  148  shew  such  instruments. 

In  Fig.  148  there  are  two  magnets  so  that  the  system  is 
astatic,  and  a  second  set  of  coils  is  added  surrounding  the 
upper  magnet.  The  current  passes  in  opposite  directions 
round  the  two  magnets,  thus  each  magnet  is  subject  to  a 
considerable  couple  and  these  two  couples  tend  to  turn  the 
system  in  the  same  direction. 

The  deflexion  of  the  needle  can  be  determined  by  ob- 
serving the  motion  of  the  image  reflected  from  the  mirror. 

Various  arrangements  are  adopted  in  practice.  Light 
from  a  lamp  traverses  a  vertical  slit  and  falls  on  the  mirror. 
The  scale  is  placed  above  the  slit  in  a  horizontal  position  ; 
the  mirror  may  be  either  concave  or  plane.  If  it  be  concave 
the  slit  and  scale  are  placed  at  a  distance  from  the  mirror 
equal  to  its  radius,  and  the  scale  is  adjusted  until  the  centre 


155] 


MEASUREMENT   OF   CURRENT 


235 


of  the  mirror  is  vertically  above  the  slit  and  halfway  between 
its  centre  and  the  scale.  A  real  image  of  the  slit  is  thus 
formed  on  the  scale.  If  the  mirror  be  fla,t  a  convex  lens  is  used. 
This  is  sometimes  placed  close  to  the  mirror  so  that  both  the 
incident  and  reflected  beams  traverse  it.  The  lens  should  have 
a  somewhat  long  focal  length  (say  1  metre)  and  its  distance 
from  the  slit  is  equal  to  its  focal  length.  The  rays  from,  the 


Fig.  148. 

slit  fall  on  the  plane  mirror  as  a  parallel  pencil  and  are  re- 
flected as  such  back  to  the  lens ;  after  traversing  it  they  form 
a  real  image  of  the  slit  on  the  scale. 


236  ELECTRICITY  [CH.  XV 

If  the  light  only  traverses  the  lens  once,  it  must  be 
arranged  as  described  in  Section  96  to  form  a  real  image  as  far 
behind  the  mirror  as  the  slit  is  in  front.  The  light  on  its  way 
to  form  this  image  is  reflected  by  the  mirror  and  the  image 
is  formed  on  the  scale. 

Instead  of  reflecting  a  beam  of  light  on  to  a  scale  we  may 
place  a  scale  in  front  of  the  mirror,  illuminate  it  suitably 
and  then  view  with  a  telescope  its  reflected  image.  Another 
modification  of  this  plan  is  to  use  a  low  power  microscope 
in  place  of  the  telescope,  attaching  a  fine  scale  just  below  the 
object  glass  of  the  microscope  and  viewing  its  reflected  image. 

Various  forms  of  sensitive  galvanometers  are  illustrated  in 
Figs.  146—148. 

Fig.  146  shews  an  ordinary  astatic  instrument,  Fig.  147 
a  simple  mirror  galvanometer,  and  Fig.  148  a  mirror  galva- 
nometer with  two  sets  of  coils  and  an  astatic  system  of 
magnets. 

Each  of  the  last  two  instruments  is  fitted  with  a  control 
magnet  to  bring  the  spot  of  light  on  to  the  scale  and  to  vary 
the  sensitiveness. 

There  are  various  other  forms  of  a  galvanometer  and  an 
account  of  some  of  these  is  given  in  Sections  220 — 222. 

It  should  be  noted  that  it  is  not  possible  to  determine  with  accuracy 
by  calculation  the  magnetic  force  exerted  on  the  needle  of  a  sensitive  gal- 
vanometer by  a  given  current.  The  coils  are  too  close  to  the  magnet  and 
the  conditions  assumed  in  the  theory  of  Section  146  are  far  from  being 
satisfied.  The  galvanometer  constant  and  reduction  factor  of  such  an 
instrument  can  only  be  found  by  a  direct  electrical  experiment  made  for 
the  purpose,  employing  a  current  measured  by  some  other  method. 

156.    Law  of  Magnetic  force  due  to  a  Current. 

We  have  deduced  our  theory  from  the  assumption  that  the 
magnetic  force  due  to  a  current  i  traversing  an  arc  of  a  circle 
of  length  I  cms.  and  radius  r  cms.  at  the  centre  of  the  circle  is 
^7/r2  in  a  direction  perpendicular  to  the  plane  of  the  circle. 
Various  experiments  have  been  devised  to  illustrate  this  law. 

Thus,  in  Fig.  149,  ABC  represents  a  circular  coil  of  given 
radius,  say  5  cms.,  containing  one  turn.  DEF  is  a  second  coil 
of  twice  the  radius  having  two  turns.  The  two  coils  are 
concentric,  and  their  planes  coincide.  They  are  connected 


155-156]  MEASUREMENT    OF   CURRENT  237 

together  so  that  the  same  current  can  circulate  in  opposite 
directions    in    the   outer    and   inner    coils    respectively,    and 


Fig.  149. 

are  placed  with  their  plane  in  the  magnetic  meridian.  A 
small  magnet  attaqhed  to  a  mirror  is  suspended  at  the  common 
centre  of  the  coils  and  a  beam  of  light  is  reflected  from 
the  mirror  on  to  a  scale.  The  connexions  are  usually  such 
that  the  current  can  be  allowed  to  pass  through  either  coil 
separately. 

Pass  the  current  through  the  inner  coil ;  a  deflexion  of 
the  spot  of  light  is  observed.  Pass  it  through  the  outer  coil, 
the  spot  is  again  deflected  ;  if  the  current  be  not  altered  by 
the  change  of  connexions  it  will  be  found  that  these  two 
deflexions  are  equal. 

Now  pass  the  current  through  the  two  coils  in  series  taking 
care  that  it  circulates  in  opposite  directions  through  the  two. 
No  deflexion  will  be  observed.  Thus  the  effect  of  the  current 
in  the  single  turn  is  balanced  by  that  of  the  same  current  in 
two  turns  of  twice  the  radius.  We  may  shew  in  a  similar  way 


238 


ELECTRICITY 


[CH.  XV 


that  the  effect  of  three  turns  of  three  times  the  radius  is  the 
same  as  that  of  the  single  turn. 

Now  when  the  radius  is  doubled  the  length  of  each  turn 
is  doubled  but  the  number  of  turns  is  also  doubled,  thus  the 
length  of  the  arc  in  which  the  current  flows  is  increased 
four-fold,  or  by  22,  so  when  the  radius  is  increased  3  times 
the  length  of  the  arc  is  increased  9  or  32  times. 

Hence  if  we  suppose  the  magnetic  force  to  vary  as  the 
length  of  the  arc,  and  this  is  reasonable,  for  each  small  element 
of  the  arc  must,  it  is  clear,  contribute  equally  to  the  force,  it 
must  also  vary  inversely  as  the  square  of  the  radius.  Thus 
the  force  will  vary  as  l/r2. 

157.  Commutators.  In  experiments  with  electric 
currents  it  is  often  desirable  to  reverse  the  direction  of  a 
current  in  one  part  of  a  circuit  without  altering  the  battery 
connexions.  This  is  done  by  means  of  a  commutator.  One 
convenient  form  of  commutator  is  shewn  in  Fig.  150. 


Fig.  150. 

A,  B,  (7,  D  are  mercury  cups  arranged  at  the  corners  of  a 
square  and  connected  to  binding  screws  as  shewn. 

The  battery  wires  are  connected  to  A  and  (7,  the  circuit  in 
which  the  current  is  to  be  reversed  to  B  and  D. 

Two  pieces  of  stout  copper  wire  or  rod  are  bent  as 
shewn  in  the  figure  and  connected  to  an  insulating  handle. 
The  four  ends  of  the  rods  fit  into  the  mercury  cups  and 
thus  A  can  be  connected  to  B  and  C  to  D,  while  by  placing 
the  rods  in  the  mercury  cups  with  their  lengths  at  right 


156-157]  MEASUREMENT   OF   CURRENT  239 

angles   to   their   original    positions   A   and  D  are  connected 
together,  and  also  B  and  C. 

Thus  if  A  be  connected  to  the  positive  pole  of  the  battery, 
in  the  first  position  the  current  flows  from  At  then  I-OK- 
external  circuit  from  B  to  D  and  thence  to  the  ba 
C,  while  in  the  second  position  the  direction  in  the 
circuit  is  from  D  to  B. 

Another  form  of  commutator  is  shewn  in  Fig.  151. 


Fig.  151. 

A  cylindrical  shaped  piece  of  ebonite,  which  can  be  turned 
round  its  axis,  carries  two  strips  of  brass  P,  Q  at  oppo- 
site extremities  of  a  diameter  of  the  cylinder.  These  are 
insulated  from  each  other,  but  P  is  connected  to  a  binding 
screw  A  by  means  of  one  of  the  supports  in  which  the  axis 
of  the  cylinder  turns,  while  Q  is  connected  to  another  screw 
C  b}7  means  of  the  second  support.  These  two  supports  are 
insulated  from  each  other. 

Two  strips  of  stiff  brass  R  and  S  connected  to  two  other 
binding  screws  B  and  D  press  against  the  cylinder.  If  the 
cylinder  were  turned  90°  in  the  right  direction  from  the 
position  shewn  P  would  be  in  contact  with  R,  Q  with  S. 
A  current  then  entering  at  A  passes  to  J5,  round  the  external 
circuit  to  D  and  then  by  the  strips  #  and  Q  to  ('.  If  the 
cylinder  be  turned  through  180°  so  that  P  is  in  contact  with 
*S',  Q  with  R,  the  current  entering  at  A  passes  through  Py  S, 
to  D  then  round  the  external  circuit  from  D  to  B  and  finally 
back  by  R  and  Q  to  C. 

In  addition  to  a  commutator  various  other  forms  of  keys 
and    switches   will   be    of   service  in  electrical  exj 
the  mode  of  action  of  most  of  these  will  however 
from  inspection. 


CHAPTER  XVI. 

MEASUREMENT  OF  RESISTANCE  AND 
ELECTROMOTIVE  FORCE. 

158.  Resistance  Boxes.  We  have  stated  already 
that  the  resistance  of  a  conductor  is  a  physical  property  of 
the  conductor  which  remains  constant  so  long  as  the 
conductor  retains  the  same  physical  conditions.  A  piece 
of  wire  therefore  has  a  definite  resistance  and  standards  whose 
resistances  are  multiples  or  sub-multiples  of  the  ohm  can  be 
constructed  out  of  Coils  of  insulated  wire.  It  is  found  by  ex- 
periment that  the  specific  resistances  of  certain  alloys  such  as 
Platinum-Silver,  Platinoid,  German  Silver,  "  Manganin "  (an 
alloy  of  Copper,  Nickel'and  Manganese)  are  much  higher  than 
those  of  the  pure  metals  and  that  their  resistance  changes  less 
with  ternperatupe  than  does  that  of  the  pure  metals.  These 
substances  therefore  are  chosen  for  resistance  coils.  Platinum- 
silver,  an  alloy  of  67  parts  of  platinum  and  33  of  silver,  is  used 
for  standard  coils  because  of  its  permanence. 

The  coils  are  either  wound  on  single  bobbins,  the  ends  of 
the  wires  being  connected  to  binding  screws,  or  are  made  up 
into  resistance  boxes. 

In  any  case  the  wipe'  is  bent  back  on  itself  at  its  middle 
and  wound  double  on  the  bobbin.  This  is  to  avoid  the  effect 
of  self-induction1  and  to  prevent  direct  magnetic  action  on 
any  galvanometer-magnet,  or  similar  instrument  in  the  neigh- 
bourhood. 

1  See  §  232. 


158]  MEASUREMENT  OF   RESISTANCE  241 

The  top  of  a  resistance  box  is  made  of  a  non-conducting 
material  and  to  this  are  attached  a  number  of  stout  brass  pieces 
as  shewn  at  A,  B,  C,  I)  in  Fig.  152.  A  small  space  is  left  between 
each  of  these  pieces  of  brass  and  the  ends  of  these  pieces  are 
ground  in  such  a  way  that  a  taper  plug  of  brass  can  be 
inserted  in  them,  and  so  put  the  brass  blocks  into  electrical 
communication. 


Fig.  152. 

Binding  screws  are  attached  to  the  two  end  blocks.  When 
the  plugs  are  all  inserted  a  current  can  pass  from  end  to  end 
through  the  blocks  and  plugs.  The  two  ends  of  each  coil  are 
soldered  to  two  consecutive  blocks,  so  that  when  a  plug  such 
as  P  is  removed,  the  current  can  pass  from  one  block  A  to  the 
next  B  only  by  traversing  the  coil.  The  resistance  of  the  coil 
is  thus  introduced  into  the  circuit  between  the  binding  screws. 

The  resistance  of  the  brass  blocks  and  plugs  is  so  small  as  to 
be  negligible  for  most  purposes,  provided  the  plugs  tit  properly, 
hence  the  total  resistance  between  the  binding  screws  is,  in  the 
case  we  have  supposed,  that  of  the  coil  connecting  A  and  B. 

The  coils  in  a  box  are  usually  arranged  thus : 

1225 
10  10  20  50 

100         100         200    •     500  units. 

Thus  with  twelve  coils  as  shewn  any  resistance  between 
0  and  1000  ohms  can  be  inserted  in  the  circuit. 

In  some  boxes  the  coils  are  arranged  in  powers  of  2  thus, 
1,  2,  2?,  23...  units.  Fewer  coils  are  needed  to  make  up  a 
given  resistance  on  this  system  than  on  the  other. 

E.  16 


242  ELECTRICITY  [CH.  XVI 

Another  arrangement  is  the  dial  box,  the  top    of    which 
is  shewn  in  Fig.  153. 


Fig.  153. 

A  series  of  eleven  brass  studs  are  arranged  in  an  arc  of 
a  circle.  The  consecutive  studs  are  connected  by  a  series  of 
10  equal  resistance  coils  and  by  means  of  a  plug  each  can  be 
connected  in  turn  to  a  block  at  the  centre  ojf  the  circle.  The 
connexions  to  the  rest  of  the  circuit  are  made  at  the  centre 
of  the  circle  and  at  the  first  stud. 

"When  the  plug  connects  the  first  stud  to  the  centre  block 
the  current  passes  through  it  directly  and  the  resistance  of 
the  arm  is  so  small  as  to  be  negligible. 

If  the  plug  is  placed  in  the  second  hole  the  current 
traverses  the  first  resistance  in  passing  through  the  box,  if  it 
be  in  the  third  hole  two  resistance  coils  are  traversed  by  the 
current,  and  so  on.  Thus  we  can  insert  in  the  circuit  any 
resistance  between  0  and  10  units.  A  second  dial,  each  coil  of 
which  is  ten  times  that  of  the  coils  of  the  first  dial,  enables 
us  to  deal  with  resistances  between  10  and  100  units,  and 
so  on. 

159.  Shunts.  It  may  often  happen  that  a  current 
which  it  is  desired  to  measure  produces  too  large  a  deflexion  in 
a  sensitive  galvanometer.  An  arrangement  whereby  a  definite 
fraction  of  the  current,  1/10  or  1/100  of  the  whole,  may  be  sent 
through  the  galvanometer  is  therefore  convenient  and  this 
is  secured  by  the  use  of  a  shunt. 


158-159]  MEASUREMENT   OF   RESISTANCE 


243 


A  shunt  is  merely  a  resistance  of  suitable  amount  which  is 
connected  across  the  terminals  of  the  galvanometer. 

Thus  let  A,  B,  figure  154,  be  the  terminals  of  the  gal- 
vanometer and  let  R  be  its  resistance,  let  A  and  B  be 
connected  through  a  resistance  *Sf.  A  current  entering  at 


Fig.  154. 

A  has  two  paths  to  B  and  is  divided  in  proportion  to  the 
conductivities  of  the  paths.  •  Let  C  be  the  whole  current, 
Cg  the  part  which  passes  through  the  galvanometer,  Cs  that 
which  traverses  the  shunt. 

Then  C  =  Cs  +  Cg. 

Also  Ca.R  =  CgS;    for  they  both   measure  the  potential 
difference  between  A   and  B. 

Thus  Ca       S° 


RC 
*~  S  +  ti' 

These  results  of  course  follow  at  once  from   the  formula 
of  §  142. 

Hence  we  have         C  —          .,         . 

o 

Thus  if  we  know  S  and  R  and  measure  Cff  we  can  find  C. 
Tn  practice  £  is  taken    as    some    definite   fraction    of   R. 
Thus  if  8JR  =  1/9  we  have  v 


16—2 


244 


ELECTRICITY 


[CH.  XVI 


or  if  SIR  =  1/99,  then 

(7=100(7,. 

A  shunt  box,  constructed  for  a  given  galvanometer, 
usually  contains  three  coils  whose  resist- 
ances are  1/9,  1/99,  1/999  of  the  resistance 
of  the  galvanometer.  One  end  of  each 
coil  (Fig.  155)  is  connected  with  one 
terminal  of  the  box.  The  other  end  of 
each  goes  to  one  of  three  insulated  brass 
blocks  attached  to  the  top  of  the  box.  Each 
of  these  blocks  can  be  connected  by  means 
of  a  plug  to  a  fourth  brass  block  which 
forms  the  other  terminal.  Suppose  the 
plug  placed  so  as  to  connect  up  the  1/99 
coil.  The  terminals  of  the  box  are  con- 
nected to  those  of  the  galvanometer  and 
99/100  of  any  current  passes  through  the  box,  1/100  through 
the  galvanometer. 

Another  form  of  shunt  box  which  can  be  used  with 
different  galvanometers  is  illustrated  in  figure  156.  This  was 
designed  by  Prof.  Ayrton  and  Mr  Mather. 


Fig.  155. 


Fig.  156. 
Let  A ,  B  be  the  terminals  of  a  galvanometer  of  resistance  7?, 


159-160]  MEASUREMENT   OF   RESISTANCE  245 

let  them  be  joined  by  a  shunt  of  resistance  S  and  let  P  be  a 
point  on  the  shunt  circuit,  the  position  of  which  can  be  varied. 
Let  the  resistance  of  AP  be  X,  and  let  a  current  be  introduced 
at  A  and  withdrawn  at  P. 

The    fraction    of   this    current   which    will    traverse   the 
galvanometer    is    Xj(R  +  S).     That   traversing   the    shunt  is 


By  giving  X  the  values  S,  S/10,  S/100,  S/1000  in  turn,  the 
fractions  of  the  current  traversing  the  galvanometer  will  be 
respectively 

S          I       S  l         S  1          S 


X  +  ti'    10  £  +  S'    100       +  £'    1000^  +  ^' 
assuming  the  current  entering  at  A  to  remain  constant. 

Thus  if  the  coil  S  be  subdivided  into  four  parts  of 
1/1000,  9/1000,  9/100  and  9/10  of  the  whole,  and  connected  as 
shewn  in  Fig.  156  so  that  the  block  P  can  be  connected  by 
means  of  a  plug  to  any  of  the  divisions,  the  arrangement  will 
serve  as  a  shunt  to  any  galvanometer. 

160.     Experiments    on    Electric    Currents.     We 

turn  now  to  the  methods  of  measuring  the  various  electrical 
quantities,  current,  electromotive  force,  and  resistance,  with 
which  we  have  been  dealing. 

Current  and  electromotive  force  are  measured  practically 
by  some  of  the  numerous  forms  of  direct  reading  ammeters 
and  voltmeters  which  are  graduated  to  give  the  amperes  or 
the  volts,  as  the  case  may  be,  directly.  Leaving  these  for 
the  present1  we  will  consider  some  experiments  in  electrical 
measurement  which  illustrate  the  various  fundamental  laws 
under  discussion. 

For  the  current  measurements  a  tangent  galvanometer  is 
required. 

An  instrument  with  a  coil  about  10  -5  centimetres  in 
radius  will  be  found  useful.  The  groove  should  be  wound 
with  three  separate  coils,  one  of  low  resistance,  containing 
3  turns,  a  second  of  about  60  turns  with  a  resistance 

1  See  Sections  220—223. 


246  ELECTKIC1TY  [CH.  XVI 

considerably  less  than  1  ohm,  the  third  with  about  600 
turns  of  thin  wire  having  a  resistance  of  200  or  250  ohms. 
The  terminals  may  be  conveniently  arranged  so  that  the  coils 
can  be  used  either  separately  or  in  series. 

We  have  seen  that  in  a  tangent  galvanometer  the  current 
in  c.G.s.  units  is  given  by  the  formula 

i  —  k  tan  0, 

where  k  the  reduction  factor  is  equal  to  Brftmr^  i  is  the 
current  and  0  the  deflexion. 

Putting  in  the  values 

#=•18,  r-10-5,  n  =  3,  Tr-3'14, 
we  find  k  —  •!. 

Thus  the  values  of  k  for  the  three  separate  coils  are 
•1,  -005,  and  '0005,  or  measuring  in  amperes  they  are  1,  '05, 
and  '005 ;  in  other  words,  deflexions  of  45°  are  caused  re- 
spectively by  currents  of  1,  '05,  and  '005  amperes. 

For  some  of  the  experiments  a  mirror  galvanometer,  §  155, 
is  required,  while  in  some  methods  of  comparing  electromotive 
forces,  and  in  verifying  Ohm's  law,  a  quadrant  electrometer 
will  be  useful. 

For  reducing  the  results  of  the  experiments  a  table  of 
tangents  will  be  required. 

TABLE  OF  TANGENTS  OF  ANGLES  FROM  0°  TO  90°. 

Angle  Tangent  Angle  Tangent  Angle  Tangent 

0°  -000  13°  -231  26°  '488 

1  -018  14  -249  27  "510 

2  -035  15  -268  28  '532 

3  -052  16  -287  29  '554 

4  -070  17  '306  30  '577 

5  -088  18  -325  31  '601 

6  -105  19  '344  32  -625 

7  -123  20  -364  33  '649 

8  -141  21  -384  34  "675 

9  -158  22  -404  35  '700 

10  -176  23  -425  36  '727 

11  -194  24  -445  37  '754 

12  -213  25  -466  38  "781 


160-161]  MEASUREMENT   OF   RESISTANCE  247 

Angle  Tangent  Angle  Tangent  Angle  Tangent 

39°  -810  56°  1-48  73°  3-27 

40  -839  57  1-54  74  3-49 

41  -869  58  1-60  75  3  -?3 

42  -900  59  1-66  76  4-01 

43  -933  60  1-73  77  4'33 

44  -966  61  1-80  78  4-71 

45  1-00  62  1-88  79  5  -15 

46  1-04  63  1-96  80  5  -67 

47  1-07  64  2-05  81  6  -31 

48  1-11  05  2-15  82  7  '12 

49  1-15  66  2-25  83  8-14 

50  1-19  67  2-36  84  9  "51 

51  1-24  68  2-48  85  11-4 

52  1-28  69  2-61  86  14-3 

53  1-33  70  2-75  87  19'1 

54  1-38  71  2-90  88  28'6 

55  1-43  72  3-08  89  57  "3 

161.     Absolute  measurement  of  a  current. 

EXPERIMENT  34.     To  measure  absolutely1  the  current  in  a 

wire. 

For  this  purpose  we  need  a  galvanometer,  the  reduction 
factor  of  which  we  can  determine  by  measurement  ;  if  the 
current  be  suitable  we  may  employ  the  tangent  galvanometer 
just  described,  using  the  coil  of  three  turns  if  we  can  measure 
their  diameter  with  sufficient  accuracy.  This  may  be  done 
either  witli  a  pair  of  callipers2,  or  by  the  aid  of  a  steel  tape 
which  is  stretched  round  the  coil  in  contact  with  the  wire. 

Measure  the  diameter  of  the  coil  in  centimetres  and  count 
the  number  of  turns,  then  calculate  the  reduction  factor  from 
the  formula  k  =  IJr/'2mr  using  the  value  '18  for  H. 

Set  up  the  galvanometer  as  described  in  Section  154  and 
connect  its  terminals  to  two  binding  screws  of  a  commutator 
or  reversing  key,  Section  157.  Connect  the  other  two  binding 
screws  of  the  key  to  the  battery  or  source  of  current,  and 
observe  the  deflexion  015  reverse  the  key  and  again  observe 
the  deflexion  02.  Then  the  current  is  given  by  the  expression 

c  =  k  tan  1  (0,  +  02). 


1  By  the  absolute  measurement  of  a  quantity  is  meant  its  determina- 
tion in  terms  of  the  fundamental  units  of  mass,  length,  and  time. 
'J  Glazebrook's  Mechanics. 


248 


ELECTRICITY 


[CH.  XVI 


The  current  so  found  is  in  c.G.s.  units.  To  obtain  the 
value  in  amperes  multiply  by  10,  for  one  c.G.s.  unit  contains 
ten  amperes. 

If  the  current  is  too  small  to  be  measured  with  the 
instrument  described  it  may  be  desirable  to  use  a  mirror- 
galvanometer.  A  convenient  instrument  is  formed  by  turning 
a  small  circular  groove  some  20  cm.  in  radius  in  a  flat  piece  of 
wood,  the  groove  being  of  such  a  size  that  a  stout  copper  wire 
will  just  lie  evenly  in  it  to  form  the  coil  of  the  galvanometer. 
The  wire  forms  as  nearly  as  possible  a  complete  circle,  and  its 
ends  are  carried  away  at  right  angles  to  the  board,  being  kept 
as  close  together  as  is  possible ;  if  the  wire  is  insulated  the 
two  ends  are  twisted  together.  This  is  to  prevent  direct 
action  on  the  magnet.  The  board  can  stand  in  a  vertical 
position.  A  hole  of  the  form  shewn  in  Fig.  157  is  cut 


Fig.  157. 

through  the  wood  at  the  centre  of  the  circle,  and  closed 
with  two  slips  of  glass,  thus  forming  a  small  cell.  In  this 
cell  a  mirror  is  suspended ;  the  mirror  has  some  small  magnets 


161]  MEASUREMENT   OF   RESISTANCE  249 

attached  to  its  back.  The  board  is  placed  in  the  magnetic 
meridian  by  the  aid  of  a  long  magnet  mounted  as  a  compass- 
needle  or  in  some  other  manner. 

A  lamp,  slit  and  scale  are  arranged  ;  using  a  lens,  unless 
the  mirror  is  concave,  so  as  to  produce  an  image  of  the  slit  on 
the  scale.  This  image  should  be  above  the  slit. 

Instead  of  the  lamp  and  slit  an  incandescent  lamp  may 
often  be  employed  witli  advantage  ;  a  lamp  with  a  straight 
filament  is  chosen,  and  a  diaphragm  arranged  to  cut  off  all 
the  light  except  that  from  a  straight  bit  of  the  filament. 

The  scale  should  lie  in  the  magnetic  meridian  parallel 
therefore  to  the  coil  of  the  galvanometer  ;  this  is  secured  by 
turning  it  round  until  two  points  on  it,  equidistant  from  the 
slit,  are  equidistant  from  the  magnet. 

Measure  the  distance  of  the  scale  from  the  magnet;  let 
it  be  a  centimetres.  Connect  the  two  terminals  of  the 
galvanometer  through  the  reversing  key  to  a  battery,  and 
note  the  deflexion  of  the  spot;  let  it  be  dl  centimetres. 
Reverse  the  current  and  observe  the  deflexion  in  the  opposite 
direction,  d%  centimetres.  If  the  adjustments  are  perfect, 
dl  will  be  equal  to  c?2,  if  the  two  be  not  exactly  equal  the 
mean  d  —  equal  to  |  (dl  +  d2)  —  will  be  free  from  some  small 
errors. 

If  0  be  the  deflexion  of  the  magnet,  since  the  reflected 
light  is  deflected  through  twice  the  angle  through  which  the 
mirror  is  turned,  we  have 

tan  20  =  -, 
a 

and  if  d  is  small  compared  with  a  we  have  approximately 

tan  (9  =  J-. 

Hr 

But  i  =    —  tan  6 


Hr      d_ 
=  2^  X  2a  ' 

The   value  of  r  is  found  by  measuring  the  diameter  of 
the  groove  in  which  the  wire  lies  with  a  finely  divided  scale 


250  ELECTRICITY  [CH.  XVI 

or  pair  of  callipers  and  correcting  for  the  thickness  of  the 
wire.  The  value  of  H  may  be  taken  in  England  as  "180  units, 
hence,  as  all  the  quantities  involved  are  known,  the  current 
is  found. 

Example.  The  radius  of  a  single  coil  galvanometer  is  20-5  cms.  ; 
the  distance  of  the  mirror  from  the  scale  is  100  cm.,  and  a  given  battery 
when  connected  to  the  circuit  produces  a  mean  deflexion  of  15-5  cm. 
Find  the  current. 

If  6  is  the  deflexion  of  the  magnet 


The  value  of  HT^TT  is  -585. 

Hence  the  value  of  the  current  is  -0454  c.o.s.  units,  or  -454  ampere. 

162.  Determination  of  Electro-chemical  Equi- 
valents. 

EXPERIMENT  35.  To  find  the  electro-chemical  equivalent 
of  copper. 

Connect*up  a  galvanometer  of  known  reduction  factor  with 
a  reversing  key,  two  or  three  cells  of  constant  E.M.F.  and  a 
voltameter. 

The  voltameter  consists  of  two  copper  plates  supported  in 
a  piece  of  wood  or  ebonite  so  as  to  hang  vertically  and  parallel 
to  each  other  in  a  beaker  containing  a  saturated  solution  of 
copper  sulphate  slightly  acid. 

Binding  screws  are  attached  to  the  plates  so  that  the 
voltameter  can  be  easily  inserted  in  the  circuit. 

Clean  the  plates  well  with  emery  paper  and  weigh  care- 
fully one  plate  which  is  to  serve  as  the  kathode  for  the 
deposition  of  the  copper.  Let  its  mass  be  W  grammes. 
Connect  this  plate  to  the  zinc  pole  of  the  battery.  Connect 
the  other  plate  of  the  voltameter  to  one  binding  screw  of 
the  commutator.  Connect  the  copper  plate  of  the  battery 
to  another  binding  screw  of  the  commutator.  Connect  the 
other  screws  of  the  commutator  to  the  terminals  of  the 
galvanometer.  Thus  with  the  key  in  either  position  the 
current  passes  through  the  cell  in  the  same  direction,  but 
is  reversed  in  the  galvanometer. 

Make  contact  with  the  key  and  allow  the  current  to  flow 
for  five  minutes,  reading  the  deflexions  of  the  galvanometer 


1G1-162]  MEASUREMENT   OF   RESISTANCE  251 

each  minute.  Reverse  the  current  in  the  galvanometer  and 
allow  it  to  flow  for  five  minutes ;  then  reverse  again  and 
after  a  third  interval  of  five  minutes,  reverse  a  third  time. 
Read  the  galvanometer  at  minute  intervals  throughout,  and 
at  the  end  of  twenty  minutes  break  the  circuit.  Remove  the 
kathode  plate  from  the  solution.  Wash  it  in  distilled  water, 
pour  over  it  a  little  alcohol  and  then  dry  it  quickly  in  a  current 
of  warm  air— it  should  not  be  put  in  a  flame.  Weigh  the  plate 
again.  Let  its  mass  be  W  grammes. 

Then  in  twenty  minutes  the  passage  of  the  current  has 
deposited  W  -  W  grammes  of  copper. 

Determine  the  mean  of  the  deflexions  from  the  readings 
found  during  the  passage  of  the  current — these  readings 
should  not  vary  greatly.  The  mean  is  found  by  adding 
together  the  various  deflexions  and  dividing  the  sum  by  the 
number  of  observations.  Let  it  be  0  and  let  k  be  the  known 
reduction  factor.  Then  the  current  is  k  tan  6.  And  this 
current  has  been  flowing  for  20  x  60  seconds.  Thus  the 
quantity  of  electricity  which  has  passed  is  20  x  60  x£tan  9. 

And  the  electro-chemical  equivalent1  being  the  ratio  of  the 
mass  of  copper  deposited  to  the  quantity  of  electricity  which 
has  passed  is  given  by 

(W  -  W)/2Q  x  60  x  k  tan  0. 

Example.  The  reduction  factor  of  the  given  galvanometer  is  •!,  the 
mean  deflexion  is  46°,  and  the  mass  of  copper  deposited  in  20  minutes  is 
•410  gramme.  Find  the  electro-chemical  equivalent  of  copper. 

The  value  of  tan  46°  is  1-036. 

Thus  the  electro -chemical  equivalent  is  equal  to 

•410/120  x  1-036 
and  this  reduces  to  -00330. 

EXPERIMENT  36.  Haviny  given  the  electro-chemical  equiva- 
lent of  copper,  to  find  the  reduction  factor  of  a  galvanometer. 

This  is  merely  the  converse  of  the  last  experiment.  -The 
arrangements  and  manipulation  are  the  same.  We  are  given 
however  that  the  electro-chemical  equivalent  of  copper  is 
•00329  gramme  per  C.G.S.  unit  of  electricity. 

1  See  Section  117. 


ELECTRICITY 


[CH.  XVI 


Now  the  current  flowing  is  the  mass  deposited  per  second 
divided  by  the  electro-chemical  equivalent. 

Hence  current  =  ^-"^JL—  . 

But  current  =  k  tan  6. 

~  20  x  60  x  -00329  x  tan  0 ' 
and  the  observations  give  us  W—  W  and  0. 

163.  Observations  on  Ohm's  Law.  According  to 
Ohm's  law  if  a  constant  current  is  traversing  a  wire  the 
difference  of  potential  between  any  two  points  is  proportional 
to  the  resistance  between  them.  Moreover  for  a  uniform  wire 
the  resistance  is  proportional  to  the  length.  Hence  in  a 
uniform  wire  carrying  a  constant  current  the  potential 
difference  between  two  points  is  proportional  to  the  distance 
between  them.  Hence  if  AB,  Fig.  158,  be  a  straight  wire 


Fig.  158. 

carrying  a  current,  L,  M,  N  a  number  of  points  on  the  wire, 
and  if  lines  LP,  MQ,  NR,  etc.  be  drawn  from  these  points 
at  right  angles  to  AB  to  represent  the  potential  differences 
between  L  and  B,  M  and  B,  N  and  B,  etc.,  then  it  is  clear 
that  since  PLJBL  =  QMjBM  =  RNJBN,  the  points  P,  Q,  R 
lie  on  a  straight  line  which  passes  through  B. 


162-163]  MEASUREMENT   OF   RESISTANCE  253 

EXPERIMENT  37.     To  verify  Ohm's  Law. 

The  experiment  may  be  carried  out  thus  : 

Some  form  of  electrometer  is  needed  to  measure  the  volts. 

A  fine  wire  of  German-silver  or  some  other  material  of 
large  specific  resistance  is  stretched  along  a  divided  scale, 
being  attached  to  two  terminal  screws  A  and  B.  A  slider 
moving  along  the  scale  carries  another  screw  C  and  a 
contact  piece  which  allows  of  contact  being  made  between 
a  wire  attached  to  G  and  any  desired  point  on  the  stretched 
wire. 

The  screw  B  is  connected  to  one  pole  of  a  constant 
battery — a  small  number  of  Daniell  cells  or  one  or  two 
storage  cells — and  to  one  pair  of  quadrants  of  a  quadrant 
electrometer.  The  screw  A  is  connected  to  the  other  pole 
of  the  battery,  while  C  is  in  connexion  with  the  other  pair 
of  quadrants  of  the  electrometer. 

The  readings  of  the  electrometer  then  give  the  difference 
of  potential  between  B  and  C.  Commence  from  B  and  make 
contact  with  the  slider  in  succession  at  a  number  of  points 
along  the  wire  noting  the  electrometer  deflexion  at  each  point. 

Then  plot  a  curve,  taking  for  abscissae  the  distances  from 
B  of  the  successive  points  of  contact,  and  for  ordinates  the 
electrometer  readings.  The  curve  will  be  found  to  be  a 
straight  line  which  passes  through  B.  Thus  the  law  is 
verified  that  when  the  current  is  constant  the  difference  of 
potential  between  two  points  is  proportional  to  the  resistance. 

Again  we  have  to  shew  that  for  a  given  resistance  the 
electromotive  force  is  proportional  to  the  current. 

To  do  this  we  may  employ  the  same  arrangements  as 
above,  but  we  include  in  the  circuit  between  the  wire  AB 
and  the  battery,  a  galvanometer  for  measuring  the  current — 
an  ammeter — and  a  resistance  box. 

Measure  with  the  electrometer  the  volts  between  A  and 
some  fixed  point  L  of  the  wire;  let  them  be  E^  Measure  also 
the  current  in  the  main  circuit;  let  it  be  ilm  Put  some 
resistance  into  the  circuit  by  taking  plugs  out  of  the  box, 
and  thus  reduce  the  current  to  i.2.  Measure  the  volts;  let 


254  ELECTRICITY  [CH.  XVI 

them  be  E2.     Continue  thus  for  various  values  of  the  currents. 
]    Then  it  will  be  found  that  we  have 


or  if  we  plot  a  curve,  taking  the  currents  as  abscissae  and  the 
volts  as  ordinates,  the  curve  will  be  a  straight  line. 

If  the  galvanometer  used   to  measure  the  current  be  a 
tangent  instrument,  and  if  the  deflexions  be  Olt  02>  ...  then 


tan  Bl      tan  <92 

where  k  is  the  reduction  factor. 
Hence  the  formulae  become 

Z/*  Ti* 

&\     _  -   •&*    _ 
tan  0!  ~  tan  02 

And  we  take  as  abscissae  of  the  curve  the  values  of  tan  6. 
These  two  experiments  completely  verify  Ohm's  law. 

164.  Voltmeters.  A  voltmeter  is  a  galvanometer 
which  is  arranged  to  measure  volts. 

Suppose  L,  M  are  two  points  on  a  wire  carrying  a  current ; 
there  is  a  certain  difference  of  potential,  E  volts  say,  between 
the  two.  If  L  and  M  be  now  connected  by  wires  to  a 
galvanometer,  the  potential  difference  is  reduced  and  some 
of  the  current  flows  through  the  galvanometer;  if  however 
the  resistance  of  this  circuit  is  very  great  compared  with 
that  of  LM,  the  potential  difference  .between  L  and  M  will 
be  altered  by  a  very  small  amount,  we  may  still  take  it  as 
E  volts,  and  the  very  small  current  through  the  galvanometer 
will  be  proportional  to  E.  By  using  a  sensitive  galvanometer 
we  may  measure  this  current  with  accuracy,  and  hence,  if 
we  know  the  resistance  in  the  galvanometer  circuit,  determine 
by  the  use  of  Ohm's  law  the  potential  difference  between  L 
and  M. 

Thus  for  example  if  the  resistance  of  the  galvanometer 
circuit  be  1000  ohms,  and  if  a  current  of  -001  ampere  is 
observed,  the  potential  difference  is  -001  x  1000  or  1  volt. 


163-165]  MEASUREMENT   OF   RESISTANCE  255 

A  galvanometer  so  arranged  is  called  a  voltmeter. 

165.     Experiments  on  Batteries. 

EXPERIMENT  38.  To  compare  the  electromotive  forces  of 
different  batteries. 

(1)  With  the  tangent  galvanometer.  Adjust  the  galvano- 
meter as  already  described  and  connect  each  battery  in  turn  to 
the  long  coil,  using  the  reversing  key.  The  resistance  of  most 
ordinary  batteries  will  be  small  compared  with  the  250  ohms 
of  the  long  coil  so  that  we  may  assume  without  great  error 
that  the  various  batteries  in  turn  are  working  through  the 
same  resistance.  Thus  the  currents  produced  are  proportional1 
to  the  electromotive  forces  and  the  currents  are  proportional 
to  the  tangents  of  the  deflexions  ;  hence  if  E^,  E*  be  the  E.M.F.'S 
of  two  batteries,  0lt  #2  the  deflexions,  then 


hence 


tan 


Example.  When  a  Daniell  cell  is  connected  to  the  galvanometer  the 
mean  deflexion  is  39°,  when  a  Leclanche  cell  is  connected  it  is  47'30°. 
Compare  the  electromotive  forces. 

We  have 

E.M.F.  Leclanche  _  tan4_7^30  _  3/09  _ 

:  '  "  "  "        ~ 


E.M.F.  Daniell  tanSS     "  -81 

If  we  take  the  E.M.F.  of  the  Daniell  -as  1-07  volts,  we  find  for  that  of  the 
Leclanch^  1-43  volts. 

There  are  various  ways  in  which  we  can  allow  for  the 
effect  of  the  battery  resistance. 

Thus  arrange  the  two  cells  in  series  joining  the  negative 
pole  of  one  to  the  positive  pole  of  the  second,  and  then 
connecting  to  the  galvanometer  the  positive  of  the  first  and 
the  negative  of  the  second.  The  E.M.F.  will  be  E1  +  EZ9  the 
sum  of  the  E.M.F.'S  of  the  two;  the  resistance  will  be 

1  In  reality  i  =  EI(B+R),  where  P  is  the  battery  resistance,  R  that  of 
the  rest  of  the  circuit.  If  K  is  small  compared  with  R  we  may  write 
without  great  error  i  =  E/R,  and  R  is  the  same  for  all  the  batteries. 


256  ELECTRICITY  [CH.  XVI 

R  +  BI  +  £2t  R  being  the  external  resistance,  Hlt  £.2  the 
battery  resistances.  Observe  the  deflexion  reversing  the 
current  as  usual,  let  the  mean  deflexion  be  0. 

Now  join  the  two  negative  poles  together  and  connect 
the  two  positives  to  the  galvanometer.  The  cells  oppose 
each  other,  thus  the  E.M.F.  is  El-E%,  but  the  resistance  is 
R  +  Bl  +  J?2  as  before.  Let  the  mean  deflexion  be  6'.  Then 
the  resistances  being  the  same  the  currents  are  proportional 
to  the  electromotive  forces.  Hence 

E^  +  EZ  _  tan  0 


Example.    With  the  same  two  batteries  as  in  the  previous  example  it  is 
found  that  0  =  52°,  0'  =  15°  30'. 

El  +  E2_     tan  62°     _  1-88 
£^B~2~tanl5°30'~  -28  * 
From  this  will  be  found  that 


Orif  JE3= 1-07  volts, 

E1  =  1-44  volts. 

The  difficulty  about  the  method  is  that  if  El  and  E2  are  nearly  equal 
then  6'  is  very  small  and  a  small  error  in  the  angle  makes  a  considerable 
error  in  the  result. 

(2)  With  a  mirror  galvanometer.  The  theory  is  the  same 
but  (a)  for  small  deflexions  we  may  assume  the  current  to  be 
proportional  to  the  deflexion,  and  (/?)  a  large  resistance  will  of 
necessity  be  inserted  in  the  circuit  to  render  the  current 
sufficiently  small  to  be  measured  on  a  sensitive  instrument, 
and  this  large  resistance  will  diminish  the  error  due  to  the 
omission  of  the  battery  resistance. 

Adjust  the  galvanometer  and  bring  the  spot  of  light  to 
the  centre  of  the  scale.  Connect  up  in  series  a  resistance  box, 
the  battery  and  the  galvanometer. 

Take  sufficient  resistance  out  of  the  box  to  give  some 
convenient  deflexion  6\.  If  a  Daniell  cell  be  used  as  the 
standard,  a  deflexion  of  107  scale  divisions  will  be  a  suitable 
value  for  8l.  Replace  the  Daniell  by  the  battery  to  be  tested 
and  read  the  deflexion  82.  Then  if  we  may  neglect  the  battery 
resistance,  and  when  several  thousand  ohms  resistance  are 


165]  MEASUREMENT   OF   RESISTANCE  257 

included   in  the   circuit    this    may   usually   be  done   without 
sensible  error,  we  have 

ET  SJ 

&2  =  03 

JL\     V 

If  we  have  made  8j=  107  for  the  Daniell  for  which  JSl  =  l'Q7 
volts,  then  clearly  E2  =  82/100  volts. 

This  method  is  sometimes  varied  by  adjusting  the  re- 
sistance in  the  circuit  until  the  deflexions  due  to  the  two  cells 
are  equal. 

We  know  then  that  the  currents  are  equal,  and  this  too 
without  making  any  assumption  as  to  the  relation  between 
current  and  deflexion.  We  have  then  that  the  electromotive 
forces  are  proportional  to  the  whole  resistance  in  circuit  in 
the  two  cases. 

Let  G  be  the  galvanometer  resistance,  S19  B2  the  two  battery 
resistances.  Connect  the  one  battery  and  insert  a  resistance 
Rl  to  give  some  convenient  deflexion.  Replace  this  battery  by 
the  second  and  insert  a  resistance  R%  to  give  the  same  de- 
flexion, then  the  resistances  in  the  two  cases  are  Bl  +  G  +  R^ 
and  £a+G  +  Jt.3. 

El      Bl  +  G  +  Rl 

Hence  =r  . 

Ez     B2+G  +  R2 

In  practice  it  very  often  happens  that  £l,  B%  and  G  are  all 
very  small  compared  with  Rl  and  R2,  and  in  this  case 


(3)  By  the  Potentiometer  method.  In  Fig.  159  AB 
represents  a  long  fine  wire  1  or  2  metres  in  length  of  con- 
siderable resistance.  The  end  A  is  connected  to  the  positive 
pole  and  the  end  B  to  the  negative  of  a  constant  battery 
—  a  storage  cell  is  suitable.  Thus  a  current  is  running  from 
A  to  B  and  there  is  a  steady  drop  of  potential  along  the  wire. 
The  positive  pole  of  one  of  the  batteries  to  be  tested  is 
connected  to  A,  the  negative  pole  is  connected  through  a 
sensitive  galvanometer  G  to  a  sliding  contact  piece  P  with 
which  contact  can  be  made  anywhere  on  the  wire. 

G.    E.  17 


258  ELECTRICITY  [CH.  XVI 

Now  the  main  current  produces  a  potential  difference 
between  A  and  P  and  in  consequence  a  current  tends  to  flow 
through  the  battery  under  test  and  the  galvanometer  between 


Fig.  159. 

A  and  P.  But  the  electromotive  force  E^  of  this  battery 
tends  to  drive  a  current  round  the  same  circuit  in  the  opposite 
direction.  If  El  is  greater  than  the  drop  in  volts  between  A 
and  P  due  to  the  main  current,  the  current  in  the  galvano- 
meter will  be  from  P  to  A\  if  E-±  is  less  than  the  drop  in  volts 
the  current  will  be  from  A  to  P. 

% 

By  shifting  the  slider  a  position  can  be  found  for  P  for 
which  there  is  no  current  through  the  galvanometer.  When 
this  is  the  case  the  E.M.  F.  E^  must  be  equal  to  the  drop  in 
volts  between  A  and  P,  and  so  long  as  the  main  current  is 
constant  this  drop  in  volts  is  proportional  to  the  resistance 
and  hence  to  the  distance  AP. 

Thus  if  1\  be  the  position  so  found  we  see  that,  if  the  main 
current  is  constant,  El  is  proportional  to  AP^. 

The  first  battery  is  now  removed  and  its  place  taken  by 
the  second,  of  E.  M.  p.  Ez  \  if  another  position  P2  be  found 
for  which  there  is  no  current,  then  E^  is  proportional  to  AP2. 

EI 

Hence  s 

In  practice  the  positive  poles  of  the  two  batteries  to  be 
compared  are  permanently  connected  to  A.  Their  negative 


165]  MEASUREMENT   OF   RESISTANCE  259 

poles  are  connected  to  two  of  the  terminals  K^,  K^  of  a  two- 
way  switch.  The  common  terminal  of  the  switch  is  connected 
to  the  galvanometer  and  the  galvanometer  to  the  slide.  With 
the  switch  in  one  position  K  is  connected  to  Klt  the  first 
battery  is  in  position  and  the  point  Pl  is  found.  Then  K  is 
switched  over  to  K^  the  second  battery  is  connected  and  P2 
is  found. 

To  eliminate  any  small  change  in  the  main  current  the  key 
is  again  transferred  to  K±  and  another  position  found  for  Pl ; 
if  the  two  positions  do  not  differ  much  the  mean  of  the  two 
corresponding  lengths  is  taken  as  the  measure  of  Elt  while  Ez 
is  measured  by  AP9. 

This  method  is  known  as  Latimer  Clark's  Potentiometer 
method  of  comparing  electromotive  forces. 

As  the  standard  a  Latimer  Clark  or  a  Weston  cell  is  generally 
used.  Since  the  E.M.  F.  of  a  Clark  cell  at  15°C.  is  1-434  volts,  it 
is  convenient  to  arrange  if  possible  that  the  length  AP  for  the 
standard  should  be  1*434  metres  or  1434  millimetres.  When 
this  is  done  the  reading  in  metres  corresponding  to  any  other 
battery  gives  its  E.  M.  P.  in  volts. 

This  result  is  attained  by  inserting  a  resistance  between  B 
and  the  negative  pole  of  the  main  battery.  When  the  Clark 
is  in  circuit  the  slider  P  is  set  to  the  required  point  1434  mm. 
from  A,  and  the  main  current  adjusted  by  means  of  this 
resistance  until  the  galvanometer  is  not  deflected.  When  this 
is  the  case  the  drop  of  potential  for  each  metre  of  the  wire  is 
1  volt. 

If  the  temperature  be  not  15°C.  the  reading  for  the  Clark 
cell  must  be  set  to  represent  its  E.M.F.  at  the  time  of  the 
observation. 

In  practice  the  whole  or  a  part  of  the  wire  can  often  be 
conveniently  replaced  by  a  series  of  resistance  coils,  each  coil 
being  equal  in  resistance  to  say  100  divisions  of  the  wire. 
The  wire  need  then  only  be  100  divisions  long.  Thus  for  the 
Clark  cell  we  should  need  14  of  the  coils  and  34  divisions  of 
the  wire  ;  for  a  cell  of  voltage  1'OSO  volts  we  need  10  coils  and 
80  divisions  of  the  wire. 

17—2 


260 


ELECTRICITY 


[CH.  XVI 


166.  The  Potentiometer,  (i)  Measurement  of  current. 
The  method  can  be  employed  to  measure  a  current  in  the 
following  manner  : 

The  current  to  be  measured  is  passed  through  a  known 
resistance  R,  of  sufficient  size  to  carry  the  current.  Thus 
a  fall  of  potential  Ri  is  set  up  between  the  ends  of  the 
resistance,  i  being  the  current. 

The  end  of  this  resistance  at  which  the  current  enters  is 
connected  with  the  terminal  A  of  the  wire  (Fig.  160),  the  other 


Fig.  160. 

end  being  connected  with  the  screw  Kz  of  the  switch  shewn 
in  Fig.  1591.  Thus  the  resistance  carrying  the  current  takes 
the  place  of  the  battery  whose  E.  M.  F.  is  to  be  found  in  terms 
of  the  Clark  or  other  standard  cell,  and  the  fall  of  potential 
Ri  is  measured  in  the  same  manner  as  the  E.  M.F.  Hence 
if  Pj,  Pz  be  as  before  the  positions  of  the  sliding  contact 
(1)  with  the  standard  in  circuit,  (2)  with  the  resistance  and 
current  connected,  we  have 

Ri     AP 


.     E,    AP, 
Hence  ^  R  '  AP,' 

(ii)  Measurement  of  resistance.  Two  resistances  R,  S 
can  also  be  compared  by  this  means. 

A  current  i  is  passed  through  the  two  resistances  in  series 
(Fig.  161):  a  second  two-way  switch  has  its  terminal  con- 
nected to  the  terminal  A.  The  positive  ends  of  the  two 

1  In  Figs.  160  and  161  the  keys  are  not  shewn.  The  figures  indicate 
the  condition  when  the  balance  is  reached.  One  galvanometer  only  is 
used,  being  connected  as  in  Fig.  159  to  each  circuit  in  turn. 


166-167]  MEASUREMENT   OF    RESISTANCE 


261 


resistances  R,  <S  are  connected  to  the  other  terminals  of  the 
switch ;  the  negative  ends  of  R  and  S  being  connected  to  Kl 
and  K2. 

Thus  the  switches  can  be  arranged  so  that  in  one  position 
R  is  connected  to  A  and  through  the  galvanometer  to  the 
slider ;  in  the  other  R  is  out  of  circuit  and  S  takes  its  place. 


CLARK  CELL 


Fig.  161. 


Now  if  i  be  the  common  current  in  R  and  S,  in  the  one 
position  the  E.  M.  F.  in  the  slider  circuit  is  Ri,  in  the  other 
it  is  Si.  Hence  if  Plt  P2  are  the  positions  of  the  slider 


Thus  R  =  S^p- 

In  many  cases  the  wire  and  slider  may  be  usefully  replaced 
by  two  resistance  boxes.  See  Glazebrook  and  Shaw,  Practical 
Physics,  Section  W. 

In  Fig.  162  we  have  a  figure  of  the  instrument  as  now 
arranged  by  Messrs  Crompton.  The  potentiometer  coils  are 
shewn  to  the  left,  and  the  rheostat  for  adjusting  the  main 
current  to  the  right ;  the  centre  dial  takes  the  place  of  the 
keys  K^  K2  of  the  previous  description  ;  by  it  various  circuits 
in  turn  can  be  connected  to  the  galvanometer. 


167.     The  Potentiometer.   Further  applications. 


We  may  use  the  apparatus  described  in  the  previous  sections 


262  ELECTRICITY  [CH.  XVI 

to  verify  the  laws  as  to  the  effects  of  connecting  batteries  in 
series  and  in  parallel.  Thus  measure  by  one  or  other  of  the 
above  methods  the  electromotive  forces  of  two  or  more  cells. 
Then  arrange  the  cells  in  series,  the  positive  pole  of  one  cell 


©A©    ©B©    ©C©       jlRWS,   ©°©    ©E©    ©F© 

: 


Fig.  162. 

being  connected  to  the  negative  of  the  preceding  one,  and 
measure  the  E.  M.  P.  of  the  combination.  It  will  be  found  that 
this  is  the  sum  of  the  electromotive  forces  of  the  separate  cells. 

Now  arrange  the  cells  in  parallel  so  that  all  the  positive 
are  connected  together  and  all  the  negative  poles,  and  measure 
the  B.M.F.  again ;  if  the  cells  all  have  the  same  E.M.F.  the  result 
will  be  found  to  be  equal  to  the  E.  M.  F.  of  one  cell ;  if  the  cells 
differ  in  E.M.F.  the  result  will  be  intermediate  between  those 
of  the  given  cells ;  it  will  depend  on  the  resistances  and 
cannot  be  calculated  unless  these  are  known. 

168.  Batteries  in  series  and  in  parallel.  Again 
connect  each  cell  up  in  turn  first  to  the  short  coil  then  to  the 
long  coil  of  the  tangent  galvanometer  and  note  the  deflexion 
in  each  case. 

Arrange  the  cells  in  series  and  connect  them  to  the  long 
coil.  The  current  flowing  will  be  greater  than  that  due  to 
each  cell  singly,  in  fact  if  Oit  6.2,  03  be  the  deflexions  due  to  the 
single  cells,  0  that  due  to  the  cells  in  series,  then  unless  one 
of  the  cells  has  an  abnormally  high  resistance,  it  will  be  found 
that  we  have  approximately 

tan  6  =  tan  Ol  +  tan  02  +  ... 

For  if  Ely  E^  ...  be  the  electromotive  forces,  £lt  B.2  the 
resistances  of  the  batteries,  il}  £,,  ...  the  currents,  R  the 


167-168]  MEASUREMENT    OF    RESISTANCE  263 

galvanometer   resistance,  and  i  the    current  when   the   cells 
are  in  series,  we  have,  taking  the  case  of  two  cells, 


_ 


Now  with  the  long  coil  of  the  galvanometer  R  is  large 
compared  with  Bl  or  J5.2.     Hence  we  may  write  approximately 


And  since  the  currents  are  proportional  to  the  tangents  of 
the  deflexion 

tan  0  =  tan  Ol  +  tan  02, 

,  in  words,  when  the  external  resistance  is  large  compared 
with  the  battery  resistance  the  current  produced  by  a  number 
of  cells  in  series  is  very  nearly  the  sum  of  the  currents  due  to 
the  individual  cells. 

For  working  a  long  telegraph  line  a  high  voltage,  obtained 
by  using  a  large  number  of  cells,  is  required.  We  may 
compare  this  with  the  problem  of  forcing  water  through  a 
tube  of  narrow  bore.  A  pump  capable  of  exerting  great 
pressure  is  needed,  the  resistance  caused  by  the  water  having 
to  pass  through  the  valves  of  the  pump  is  small  compared 
with  that  which  arises  from  the  narrow  bore  of  the  pipe. 
Nothing  much  is  gained  by  increasing  the  passages  in  the 
pump.  The  pressure  which  it  can  exert  determines  the  flow. 

Now  connect  the  cells  still  arranged  in  series  to  the  short 
coil.  The  current  is  not  much  greater  than  that  given  by  one 
cell  alone.  The  electromotive  force  it  is  true  has  been  in- 
creased, but  the  resistance  being  chiefly  due  to  the  battery 
—the  resistance  of  the  short  thick  coil  is  very  small  compared 
with  the  resistance  of  the  cell  —  the  resistance  of  the  circuit 
has  been  increased  with  the  increase  in  E.  M,F.  and  there  is  no 
gain  in  current, 


264  ELECTRICITY  [CH.  XVI 

Taking  the  cells,  n  in  number,  as  all  equal,  the  E.  M.  F.  is 
nE  and  the  resistance  nB.     Thus  we  have 


~  nB  +  R' 

and  since  R  is  small  compared  with  B  we  may  neglect  it  in 
the  denominator  and  hence 

i  =  nEfnB  =  E\B 
=  current  due  to  one  cell  through  the  short  coil. 

Now  arrange  the  cells  in  parallel  and  connect  them  to  the 
long  coil  ;  the  current  does  not  differ  much  from  that  due  to 
a  single  cell.  The  E.  M.F.  —  assuming  the  cells  alike  —  is  the 
same  as  that  of  each  cell,  the  battery  resistance  is  reduced 
by  the  combination,  but  in  any  case  the  battery  resistance 
is  a  small  fraction  of  the  total  resistance,  thus  the  total 
resistance  is  not  appreciably  altered  and  the  current  is  nearly 
the  same  as  that  due  to  one  cell. 

Connect  the  cells,  still  arranged  in  parallel,  to  the  short 
coil.  The  current  is  considerably  greater  than  that  given  by 
a  single  cell  ;  for  the  battery  resistance  is  in  this  case  the 
main  portion  of  the  total  resistance  and  the  battery  resistance 
is  reduced  by  connecting  the  cells  in  parallel. 

To  put  the  result  in  symbols,  if  there  be  n  cells  all  alike 
connected  in  parallel,  the  E.  M.  F.  is  E  and  the  resistance  Bin. 

Thus  the  current  is  given  by 
._       E 
~ 


For  the  long  coil  Bjn  is  small  compared  with  R,  and  the 
current  is  approximately  Ej  R,  the  same  as  that  due  to  one 
cell  through  the  long  coil  ;  for  the  short  coil  R  is  small 
compared  with  JB/n  so  that  the  current  is  nEjB  or  n  times 
that  due  to  one  cell  working  through  the  short  coil. 

169.  Comparison  of  Resistances.  Ohm's  law  is 
the  basis  of  various  methods  of  comparing  resistances. 

EXPERIMENT  39.  To  compare  two  resistances  by  the  use  of 
a  tangent  galvanometer  and  resistance  box  or  coil  of  known 
resistance. 


168-169]  MEASUREMENT    OF   RESISTANCE  265 

The  coil  of  the  galvanometer  which  has  about  60  turns 
and  a  resistance  of  less  than  1  ohm  may  conveniently  be  used. 

Connect  in  series  a  constant  cell,  the  galvanometer, 
adjusted  as  already  described,  a  key,  and  the  unknown 
resistance  R.  Let  B  be  the  battery  resistance,  G  the  galvano- 
meter resistance,  and  let  Ol  be  the  deflexion.  Then  the  total 
resistance  in  circuit  is  B  +  G  +  R. 

Replace  the  unknown  resistance  by  the  standard  S  and 
let  0  be  the  deflexion.  The  total  resistance  is  B  +  G  4  •  S. 

The  currents  are  in  the  ratio  of  tan  6l  to  tan  0,  but  the 
E.  M.  F.  being  the  same  in  the  two  cases  the  currents  are 
inversely  proportional  to  the  resistances.  Hence  they  are 
also  in  the  ratio  of  B  +  G  +  S  to  B  +  G  +  R. 

E^G  +  R  _  tan_0 

'B  +  G  +  S  "tanfl/ 

Now  S  is  known  ;  thus  if  B  and  G  are  known  R  can  be 
found.  In  many  cases  we  are  sure  that  B  and  G  are  so  small 
that  we  may  neglect  them  ;  when  this  can  be  done 

R      tan  0 


tf    tan  6 

Hence  R  =  ti  .—     jr. 

tan  6l 

If  a  resistance  box  is  used  so  that  S  is  adjustable  we  can 
proceed  thus. 

Take  plugs  out  of  the  box  until  the  deflexion  in  the 
second  case  is  equal  to  that  in  the  first,  then  the  currents  are 
equal  and  the  electromotive  force  is  the  same.  Hence  the 
resistances  must  be  the  same. 

Hence  B+G  +  R  =  £+G  +  S. 

Hence  R  =  S. 

This  assumes  of  course  that  the  box  is  such  that  it  is 
possible  to  find  a  resistance  8  in  it  equal  to  the  given  re- 
sistance. Clearly  this  is  not  always  possible,  but  if  the  box 
is  divided  to  ohms  we  can  find  two  resistances  S  ohms 
and  S  +  1  ohms  between  which  R  must  lie.  This  will  often 
be  sufficiently  accurate  for  our  purposes. 


266 


ELECTRICITY 


[CH.  XVI 


To  carry  out  the  experiment  the  apparatus  may  be  ar- 
ranged as  in  Fig.  163  in  which  K,  K^,  Kz  is  a  two-way  switch. 
With  the  switch  connecting  K  and  A^  the  current  passes 


Fig.  163. 

through  j?,  the  standard  S  is  out  of  circuit ;  with  K  and  K* 
in  connexion  the  current  passes  through  S,  the  unknown 
coil  is  out  of  circuit. 

The  formula  given  above  assumes  that  a  tangent  galvano- 
meter is  used  ;  this  may  in  practice  be  replaced  with  advantage 
by  a  direct  reading  ammeter. 

EXPERIMENT  40.  To  examine  the  manner  in  which  the  re- 
sistance of  a  wire  depends  on  its  length,  sectional  area,  and 
material. 

For  this  purpose  there  are  given  two  coils,  each  marked  A, 
of  German-silver  or  some  other  material  of  high  resistance 
of  the  same  length  and  cross  section,  a  third  coil  B  of  the 
same  material  and  length  as  A  but  with  double  the  sectional 
area,  and  a  fourth  coil  C  of  iron  of  the  same  length  and 
sectional  area  as  A. 

First  place  A  in  circuit  and  observe  the  deflexion  Ol,  then 
introduce  the  second  coil  A  in  series  with  the  first,  let  the 
deflexion  be  #2.  Replace  the  two  coils  by  fl,  let  the  deflexion 
be  03 ;  finally  substitute  G  for  B  and  let  the  deflexion  be  04. 

Look  out  the  tangents  of  the  various  angles.     Then  if  we 


169]  MEASUREMENT   OF   RESISTANCE  267 

assume  the  resistances  of  the  battery  and  galvanometer  to  be 
small,  and  write  A,  B,  C  for  the  resistances  of  the  three  coils, 
we  have  seen  that  the  currents — measured  by  the  tangents  of 
the  deflexions — are  inversely  as  the  resistances, 

A  and  A  in  series      tan  6^ 
A  ~  tan  02 ' 

and  we  shall  find  that  tan  Bl  =  2  tan  02-  Thus  the  resistance 
of  two  equal  coils  in  series  is  double  that  of  either  coil. 

B      tan  Ol 


J 


and  we  shall  find  that  tan  03  =  2  tan  0lt 
Hence  B  =  \A. 

Thus  by  doubling  the  cross  section  we  halve  the  resistance  ; 
the  resistance  of  a  coil  of  given  length  is  inversely  proportional 
to  the  area  of  its  cross  section. 

Again  C  and  A  are  of  the  same  length  and  cross  section  but 
of  different  materials,  the  ratio  tan  ^/tan  04  measures  the  ratio 
of  the  specific  resistances  of  iron  and  German-silver. 

By  arranging  coils  in  series  or  iri  parallel  and  measuring 
their  resistances  we  can  verify  the  laws  given  in  Sections  139, 
140. 

Thus  let  the  deflexions  be  as  below  : 
With  a  standard  coil  $  in  circuit  9  \ 
With  A  in  circuit  015  with  C  in  circuit  04 ; 
With  A  and  C  in  series  65 ; 
With  A  and  C  in  parallel  06. 

tan  9 

Hence  A  =  b  r       .,  , 

tan  0, 

tan  #4 ' 

A  and  C  in  series  =  S—     -T.  , 
tan  05 

A  and  C  in  parallel  =  S  - — :    n  . 
tan  9R 


268  ELECTRICITY  [CH.  XVI 

On  evaluating  these  expressions  we  find  that  the  resistance 
of  A  and  C  in  series  is  equal  to  A  +  C,  while  the  resistance  R 
of  A  and  C  in  parallel  satisfies  the  relation 

i_l      I 

R~  A  +  C' 

so  that  fi 


In  most  of  the  above  experiments  it  has  been  supposed  that  the 
resistances  of  the  battery  and  galvanometer  are  negligible.  If  this 
is  not  the  case  experiments1  can  be  arranged  to  find  them,  but  these 
are  complicated  and  in  any  case  the  methods  given  in  the  following 
section  are  free  from  this  and  other  defects  of  the  methods  just  described 
which  assume  (1)  that  the  E.M.F.  of  the  battery  remains  constant,  and 
(2)  that  the  current  can  be  read  with  sufficient  accuracy  on  the  tangent 
galvanometer  or  ammeter,  if  a  direct  reading  ammeter  be  employed. 
Frequently  neither  of  these  conditions  is  satisfied. 

170.  Wheatstone's  Bridge.  Consider  an  arrange- 
ment in  which  (Fig.  164)  two  circuits  ACS,  ADB  are  open 
to  a  current  flowing  into  the  circuit  at  A  and  out  at  B. 


Fig.  164. 

The  potential  at  A  is  higher  than  that  at  B  and  falls  as 
we  pass  along  either  circuit  from  its  value  at  A  to  its  value 
at  B.  If  we  take  a  point  C  on  the  one  circuit  we  can  find  a 
point  D  on  the  other  which  has  the  same  potential  as  C. 
To  do  this  connect  a  wire  to  the  circuit  at  B  and  to  one 
terminal  of  a  galvanometer  G.  Attach  a  second  wire  to 
the  other  terminal  of  the  galvanometer,  and  make  contact 
with  the  loose  end  at  various  points  on  the  circuit  ADB. 
If  the  point  of  contact  be  near  A,  the  galvanometer  indicates 

1  Glazebrook  and  Shaw,  Practical  Physics. 


169-170}  MEASUREMENT   OF   RESISTANCE  269 

a  current  in  one  direction,  if  the  point  of  contact  be  near  B, 
the  current  in  the  galvanometer  is  in  the  opposite  direction ; 
a  position  can  be  found  for  which  there  is  no  current  in  the 
galvanometer.  Let  this  position  be  D.  Then  it  is  clear  that 
C  and  D  are  at  the  same  potential. 

Thus 

Fall  of  potential  from  A  to  C  =  Fall  of  potential  from  A  to  Z>, 
and 
Fall  of  potential  from  C  to  B  =  Fall  of  potential  from  D  to  B. 

Now  when  a  given  current  is  flowing  in  a  wire  the  fall  of 
potential  between  any  two  points  in  the  wire  is  proportional 
to  the  resistance  between  these  points. 

Thus 

resistance  from  A  to  C  _  fall  from  A  to  C 
resistance  from  C  to  £     fall  from  C  to  B 

fall  from  A  to  D  _  resistance  from  A  to  D 
fall  from  D  to  B      resistance  from  D  to  B' 

Thus  let  P  ohms  =  resistance  of  AC, 
Q  ohms  -  „  „  CB, 
R  ohms  =  „  ,,  AD, 
S  ohms  =  „  „  DB. 

Then  we  have  arrived  at  the  result  that  when  there  is  no 
current  between  C  and  D  we  must  have 

P_R 
Q~S' 

If  then  three  of  the  resistances  P,  Q  and  S  are  known  we 
can  find  the  fourth  R,  while  clearly  it  is  not  necessary  to  know 
the  actual  values  of  P  and  Q  provided  the  ratio  P/Q  is  known. 

We  may  arrive  at  the  same  result  graphically  thus  : 

Let  AC,  CB,  Fig.  165,  represent  two  resistances  P,  Q 
respectively.  Let  AL  at  right  angles  to  AB  be  the  B.M.F.  in 
the  circuit  ACS.  Join  LB,  and  draw  CM  parallel  to  AL  to 
meet  LB  in  M  and  MN  parallel  to  AB  to  meet  AL  in  N. 


270 


ELECTRICITY 


[CH.  XVI 


Then  clearly  CM  measures  the  potential  difference  between 
C  and  B. 


M' 


Fig.  165. 

Again  let  A' D  represent  R  and  DB'  S,  D  being  the  point 
on  A  DB  which  is  at  the  same  potential  as  (?,  and  construct  a 
similar  figure  for  the  circuit  A  DB.  It  is  clear  from  the  figure 
that  there  is  such  a  point,  for  A  and  A'  are  at  the  same  potential 
as  also  are  B  and  R.  Then  since  the  E.  M.  F.  between  A  and 
B  is  the  same  along  either  path  we  have  AL  —  A'L\  and  since 
the  E.  M.  F.  between  C  and  B  is  the  same  as  that  between  D 
and  B  we  have  CM  =  DM'. 

Hence  LN=L'N'. 

Thus  we  must  have 

P_A^_  MN 
Q  ~  CB  ~  CB 

or  as  before 


M'N'     ADR 


CB        CB 


PR 


J 


hence 


P 

Q' 


When  this  relation  is  satisfied  so  that  an  E.M.F.  in  the 
branch  AB  produces  no  current  in  the  branch  CD,  these  two 
conductors  are  said  to  be-Gonjugate  to  each  other. 

The  apparatus  for  measuring  resistance  by  this  method 
takes  various  forms. 


170] 


MEASUREMENT   OF   RESISTANCE 


271 


EXPERIMENT  41.     To  measure  a  resistance  by  Wheatstone's 
bridge  method  with  a  wire  bridge. 

A    fine    uniform  wire   is    stretched   along  a  metre   scale 
(Fig.  166).     This   constitutes  the  conductor  AOB,  a  sliding 


piece  moving  along  the  scale  permits  of  contact  being  made 
at  any  point  C  of  the  wire.  The  ends  A,  B  of  the  wire  are 
soldered  to  two  thick  plates  of  copper  each  of  which  carries 
two  binding  screws.  A  third  thick  plate  of  copper  fixed 
between  the  other  two  has  three  binding  screws  attached 
to  it.  This  constitutes  the  point  D  of  Fig.  165.  The  wire 
whose  resistance  it  is  required  to  measure  is  attached  between 
A  and  D.  Between  D  and  B  is  placed  a  known  resistance  /S. 
The  wires  from  a  battery  are  joined  to  the  apparatus  at  A 
and  B  ;  the  battery  circuit  should  also  contain  a  key.  Wires 
from  a  galvanometer,  usually  an  astatic  or  other  sensitive 
instrument,  are  joined  to  C  and  D. 

The  battery  circuit  is  closed,  and  the  slider  is  moved  along 
the  scale  making  contact  at  various  points  until  a  position 
is  found  for  which  the  galvanometer  needle  is  not  deflected. 

When  this  result  is  attained  we  know  that 


Read  the  position  of  C  on  the  scale  and  thus  measure  the 
lengths  AC  and  CB,  let  them  be  a  cm.  and  b  cm.  respectively. 


272 


ELECTRICITY 


[CH.  XVI 


Then  since  ACB  is  a  uniform  wire  the  resistance  of  -any 
portion  of  it  is  proportional  to  its  length. 

Thus  P/Q  =  a/b. 

Hence  R/S  =  a/b, 

or  ^=£T- 

o 

If  then  S  is  known  R  can  be  found. 

The  resistance  8  may  either  be  a  single  coil  of  wire  having 
a  known  value,  or  more  conveniently  a  resistance  box  out  of 
which  suitable  plugs  have  been  taken ;  the  measurements,  it 
can  be  shewn,  will  be  most  accurate  when  a  is  as  nearly  as 
possible  equal  to  b.  This  condition  can  be  approximately 
secured  if  S  is  adjustable.  A  value  is  taken  for  #  and  then  R 
is  found.  The  resistance  AS'  is  then  altered  so  as  to  be  nearly 
equal  to  this  value  of  R  and  then  the  experiment  is  repeated 
giving  a  more  accurate  value  for  R. 

EXPERIMENT  42.  To  find  the  resistance  of  a  coil  by  Wheat- 
stone's  bridge  method  using  a  Post  Office  box  of  coils. 

In  this  method  the  three  resistances  P,  Q,  S  are  taken 
from  a  resistance  box  (Fig.  167).  Each  of  the  arms  AC,  CB 


/-4I 


5000     2000  1000      »000      5OO       2OO        IOO IOO 


Fig.  167. 

of  the  bridge  usually  contains  three  coils  whose  resistances 
are  10,  100,  and  1000  ohms  respectively.  Thus  by  taking 
a  single  plug  out  of  A  C,  P  may  have  any  of  the  values  10,  1 00 


170]  MEASUREMENT   OF   RESISTANCE  273 

or  1000  ohms,  while  by  taking  a  single  plug  from  CB,  O  may 
be  either  10,  100  or  1000. 

These  coils  are  spoken  of  as  the  ratio  arms  of  the  bridge 
and  the  value  of  P/Q  may  be  either  -01,  -1,  1,  10  or  100. 

The  arm  S  or  BD  contains  a  number  of  coils  by  means  of 
which  any  resistance  between  1  and  10,000  ohms  may  be 
unplugged.  It  is  important  to  take  care  before  beginning  an 
experiment  that  all  the  plugs  are  firmly  in  their  places. 

Double  binding  screws  are  attached  to  the  box  at  A 
and  D,  and  single  screws  at  B  and  C '  \  the  box  is  generally 
so  arranged  that  by  removing  a  copper  strip  the  connexion  at 
B  between  S  and  Q  may  be  broken. 

A  sensitive  reflecting  galvanometer  is  connected  with  a  key 
in  circuit  between  B  and  D ;  while  the  battery,  also  with  a  key 
in  circuit,  is  joined  to  the  box  at  A  and  B.  The  resistance 
/?,  which  is  to  be  measured,  is  placed  between  A  and  D. 

When  a  measurement  is  to  be  made  the  sensitiveness  of 
the  galvanometer  is  reduced  either  by  means  of  a  shunt  or  by 
lowering  the  control  magnet  and  10  or  100  ohms  taken  from 
each  of  the  ratio  arms.  Thus  the  value  of  P/Q  is  unity. 

Some  resistance,  say  10  ohms,  is  then  taken  from  the  arm 
S,  the  battery  circuit  is  closed,  the  galvanometer  key  depressed 
and  the  deflexion  of  the  spot  of  light  noted ;  let  us  suppose 
that  it  is  to  the  right. 

The  value  of  S  is  then  altered,  let  us  suppose  it  is  increased 
to  20  ohms,  and  the  observation  repeated ;  if  the  deflexion  is 
still  to  the  right  but  greater  than  before  it  is  clear  that  the 
change  in  S  has  been  in  the  wrong  direction,  it  needs  to  be 
decreased ;  if  the  deflexion  is  less  than  before  though  still  to 
the  right  it  is  clear  that  S  is  still  too  small.  We  will  suppose 
the  latter  to  be  the  case ;  increase  S  again  and  proceed  thus 
until  a  deflexion  to  the  left  is  obtained.  After  some  few 
trials  we  shall  be  able  to  find  two  resistances  which  differ  by 
one  ohm,  the  smaller  of  which  gives  a  deflexion  to  the  right 
while  the  larger  gives  a  deflexion  to  the  left. 

Let  these  resistances  be  51  and  52  ohms.  Then  the 
required  value  of  /S  lies  between  these  -two,  and  since 


G.   E. 


18 


274  ELECTRICITY  [CH.  XVI 


and 

the  value  of  R  lies  between  51  and  52  ohms. 

Now  remove  the  shunt  or  raise  the  control  magnet  to 
make  the  galvanometer  more  sensitive  and  make  P  10  ohms, 
Q  100  ohms,  then  P/Q  =  l/W  and  the  value  of  S  required  to 
give  a  balance  will  be  10  times  what  it  was  previously.  It 
will  therefore  lie  between  510  and  520.  Proceeding  as  before 
we  can  find  two  resistances,  512  and  513  say,  between  which 
the  required  value  of  S  must  lie  and  since  R  is  SjW  the  value 
of  R  is  between  51*2  and  51*3  ohms. 

Proceeding  thus,  making  P  10  and  Q  1000,  so  that 

PIQ  =  1/100, 

we  can  find  two  resistances,  say  5125  and  5126  ohms,  between 
which  S  must  lie.  Thus  finally  the  value  of  R  is  between 
51-25  and  51-26  ohms. 

171.  Resistance  of  a  galvanometer.  If  a  second 
galvanometer  is  available  the  resistance  of  a  galvanometer 
coil  can  be  measured  like  that  of  any  other  wire  ;  it  is  some- 
times convenient  to  use  a  galvanometer  in  the  measurement 
of  its  own  resistance. 

EXPERIMENT  43.  To  measure  by  Thomson's  method  the 
resistance  of  a  galvanometer. 

Consider  a  Wheatstone's  bridge  arrangement  in  which  the 
conjugate  condition  is  satisfied  so  that  RjS  =  P/Q,  then  the 
battery  in  AB  produces  no  current  in  CD  :  it  follows  therefore 
that  so  far  as  the  current  in  the  other  branches  of  the  circuit 
is  concerned  it  is  immaterial  whether  CD  is  open  or  closed  ;  if 
there  be  a  key  in  CD,  nothing  in  the  rest  of  the  circuit  is 
altered  by  opening  or  closing  this  key  ;  while  conversely  we 
may  infer  that  if  the  currents  in  the  rest  of  the  circuit  are 
not  altered  by  opening  or  closing  the  circuit  CD  the  conjugate 
condition  is  satisfied  and 

*=*•%• 

To  apply  this,  Fig.  168,  the  galvanometer  whose  resistance 
is  to  be  measured  is  placed  as  the  resistance  R  in  the  arm  AD 


170-172]  MEASUREMENT   OF   RESISTANCE  275 

and  a  key  is  placed  in  CD.    The  ratio  arms  P  and  Q  are  made 

equal  and  a  resistance  taken  out 

in  the  arm   8.     On    closing  the 

battery  circuit  the  galvanometer 

is    deflected   by   the    current   in 

AD. 

It  is  usually  possible  either 
by  inserting  resistance  in  the 
battery  circuit  or  by  the  use  of 
the  control  magnet  or  of  some 
more  powerful  permanent  mag-  Fig.  168. 

net  to  bring  the  spot  on  to  the 

scale  again.  Let  this  be  done  and  note  its  position.  Now 
make  contact  with  the  key  K  in  CD ;  the  spot  will  in  general 
be  deflected,  shewing  that  the  current  in  AD  is  affected  by 
closing  the  circuit  CD,  thus  the  conjugate  condition  has 
not  been  obtained. 

By  adjusting  S  however  this  deflexion  can  be  made  small, 
and  proceeding  as  in  Experiment  42  we  can  find  two  resist- 
ances differing  by  1  ohm  for  one  of  which  there  is  a  deflexion 
to  the  left,  for  the  other  to  the  right.  Thus  R  lies  between 
these  two. 

Now  make  P/Q  equal  to  1/10  and  find  two  other  resistances 
between  which  S  lies  as  in  Experiment  42.  In  this  way  a  value 
can  be  found  for  S  which  makes  AB  and  CD  conjugate  and 
thus  R  is  equal  to  S .  P/Q. 

It  should  be  noted  that  the  condition  to  be  satisfied  in  this  case  is  not 
that  there  should  be  no  current  in  the  galvanometer  but  that  there  should 
be  no  change  in  that  current  on  depressing  the  key  in  the  branch  CD. 

Moreover  it  must  be  remembered  that  there  is  a  current  in  the 
galvanometer  which  is  changed  in  amount  by  each  alteration  of  the 
resistances  P,  Q  or  S ;  thus  any  alteration  in  the  resistances  will  cause 
a  motion  of  the  spot  and  it  may  need  a  corresponding  change  in  the 
control  magnet  to  bring  it  back  on  to  the  scale. 

It  is  desirable  in  this  experiment  that  the  battery  used  should  be  a 
fairly  constant  one,  otherwise  changes  in  its  E.M.F.  when  the  key  is  depressed 
may  produce  a  motion  of  the  spot,  although  the  conjugate  condition  is 
satisfied. 

172.    Resistance  of  a  battery- 

EXPERIMENT  44.  To  find  by  Mance's  method  the  resistance 
of  a  battery, 

18—2 


276  ELECTRICITY  [CH.  XVI 

If  we  remember  that  the  resultant  effect  of  a  number  of 
electromotive  forces  is  the  sum  of  the  effects  due  to  the 
separate  electromotive  forces  it  is  clear  that  we  may  place 
a  battery  in  the  arm  AD  of  a  Wheatstone's  bridge  without 
altering  the  conjugate  condition  that  the  current  in  CD  should 
be  independent  of  the  E.M.F.  in  AB. 

Suppose  now  that  as  shewn  in  Fig.  169  the  battery  whose 
resistance    is     required     is 
placed  in  AD  and  a  key  in 
AB,  the  galvanometer  being 
in  CD. 

The  battery  will  send  a 
current  through  CD  and  the 
galvanometer  will  be  de- 
flected, if  we  can  arrange 
the  resistances  so  that  this 
deflexion  may  be  indepen-  Fig.  169. 

dent  of  the  E.M.F.  in  AB; 
we  know  that  AB  and  CD  are  conjugate,  and  hence  that 

p 

Battery  resistance  =  R  —  S .  -^ . 

Now  when  the  conjugate  condition  is  satisfied  the  current 
in  CD  is  the  same  whatever  be  the  E.M.F.  in  AD]  take  the 
case  when  this  E.M.F.  is  zero,  C  and  D  are  connected  together 
through  the  key  by  a  wire  resistance ;  this  will  produce  no 
effect  on  the  galvanometer.  Take  again  the  case  when  the 
E.M.F.  between  A  and  B  is  exactly  the  amount  required  to 
balance  that  produced  between  these  points  by  the  current 
from  the  battery  in  AD,  so  that  there  is  no  current  in  AB ; 
still  the  effect  on  the  galvanometer  is  zero,  this  last  condition 
must  be  the  same  as  holds  when  the  connexion  between  A  and 
B  is  broken  or  the  resistance  between  these  points  made 
infinite. 

Thus  if  the  resistances  P,  Q  and  S  are  so  arranged  that 
the  galvanometer  deflexion  is  the  same  whether  the  key  in 
AB  be  depressed  or  not,  we  know  the  conjugate  condition  is 
satisfied. 

To  carry  out  this  in  practice  the  ratio  arms  are  made  equal 


172-173]  MEASUREMENT   OF   RESISTANCE  277 

and  a  resistance  taken  out  in  the  arm  S ;  the  galvanometer  is 
deflected  and  the  spot  is  brought  on  to  the  scale  by  shunting 
the  galvanometer1  if  required,  and  by  the  use  of  the  control 
magnet.  The  resistance  S  is  then  adjusted  until  the  galvano- 
meter deflexion  is  not  altered  by  making  or  breaking  the 
connexion  in  the  branch  AB  by  means  of  the  key.  As  in 
the  previous  experiments  two  values  are  found  for  S,  one  of 
which  gives  a  deflexion  to  the  right,  the  other  to  the  left, 
and  the  true  value  lies  between  these.  The  ratio  P/Q  is  then 
altered  to  1/10  and  the  first  decimal  place  in  the  value  of  R 
found  as  in  Experiment  42. 

As  in  Thomson's  method  an  alteration  of  P,  Q  or  S  alters  the 
permanent  current  through  the  galvanometer  and  the  spot  may  need 
readjustment  after  each  change  in  these  resistances. 

173.  Specific  Resistance.  The  relation  between  the 
resistance  of  a  wire  and  its  specific  resistance  or  resistivity 
has  been  found  in  Section  140. 

If  I  cm.  be  the  length,  a  sq.  cm.  the  area  of  the  cross 
section,  p  the  specific  resistance  and  R  the  resistance  of  a 
wire.  Then  we  have  seen  that 


,  ,  Ra 

and  hence  p  =  —j- . 

EXPERIMENT  45.  To  measure  by  Wheatstone's  bridge  the 
specific  resistance  of  the  material  of  a  wire. 

To  find  the  specific  resistance  of  a  wire  we  have  to 
measure  its  resistance  R  ohms,  its  length  I  cm.,  and  the 
area  of  its  cross  section  a  sq.  cm. 

The  resistance  of  the  wire  should  if  possible  be  several 
ohms,  so  as  to  render  its  accurate  measurement  possible. 
If  sufficient  wire  is  available,  cut  off  a  length  having  a 
resistance  of  3  or  4  ohms  at  least.  Measure  the  resistance 
carefully.  Measure  with  a  metro  scale  or  steel  tape  the 
length  of  the  wire ;  for  this  purpose  it  will  be  necessary 

1  Since  we  are  measuring  the  battery  resistance  it  is  clear  that  we 
must  not  shunt  the  battery. 


278  ELECTRICITY    .  [CH.  XVI 

to  lay  out  the  wire  in   a  straight  line,    but   care  must  be 
taken  not  to  stretch  it, 

Care  must  be  also  taken  to  measure  exactly  to  the  points 
in  which  the  wire  is  clamped  by  the  screws  of  the  bridge ;  in 
some  cases  the  wire  may  be  soldered  to  thick  pieces  of  copper 
of  negligible  resistance ;  these  are  connected  to  the  bridge. 

.  Measure  by  means  of  the  wire  gauge  or  in  some  other 
way  the  area  of  the  cross  section ;  this  is  most  accurately 
done  by  finding  the  weight  of  water  displaced  by  a  known 
length  of  the  wire ;  the  weight  of  water  displaced  gives  the 
volume  of  the  wire1,  and  by  dividing  the  volume  by  the 
length,  the  area  of  the  cross  section  is  found.  Then  knowing 
the  resistance,  the  length,  and  the  cross  section,  the  specific 
resistance  is  found. 

Example.  A  piece  of  wire  1  metre  in  length  is  weighed  in  air  and 
in  water  and  the  loss  of  weight  is  found  to  be  '825  gramme.  Thus 
omitting  corrections  for  the  temperature  of  the  water,  the  volume  of 
the  wire  is  -825  c.cm.,  and  the  area  of  its  cross  section  is  -00825  sq.  cm. 
A  length  of  25  metres  is  measured  off  and  is  found  to  have  a  resistance 
of  6-05  ohms.  Hence  the  specific  resistance  which  is  equal  to  Rajl  is 
•00825  x  6-05/2500,  and  this  comes  to  -00001996  ohms,  or  19*96  microhms 
for  1  c.cm. 

1  Glazebrook's  Hydrostatics  §  50,  Exp.  14. 


CHAPTER   XVII. 


MEASUREMENT   OF   QUANTITY  OF   ELECTRICITY, 
CONDENSERS. 

174.  The  Ballistic  Galvanometer.  The  quantity 
of  electricity  carried  by  a  current  is  measured  by  the  product 
of  the  current  and  the  time  of  flow.  In  some  cases  a  finite 
quantity  is  carried  round  a  circuit  in  a  very  brief  time, 
the  current,  measured  by  the  rate  of  flow,  is  enormously 
great,  but  the  time  of  flow  being  very  small  the  quantity 
carried  is  finite ;  in  such  a  case  we  can  use  a  galvanometer 
to  measure  the  quantity. 

When  a  steady  current  is  passing  through  the  coils  of  a 
galvanometer,  force  is  exerted  on  the  magnet  and  the  steady 
deflexion  of  the  magnet  becomes  a  means  of  measuring  the 
current.  The  magnet  is  deflected  until  the  force  due  to 
the  current  balances  the  force  arising  from  the  control  magnet 
or,  to  put  it  rather  differently,  until  the  rate  at  which  the 
current  tends  to  produce  momentum  balances  the  rate  at 
which  the  control  magnet  tends  to  produce  it.  The  rate  at 
which  the  current  produces  momentum  is  proportional  to  the 
current,  and  the  impulse  or  total  quantity  of  momentum 
produced  is  proportional  to  the  product  of  the  current  and 
the  time  during  which  it  has  been  flowing.  But  the  product 
of  the  current  and  the  time  measures  the  quantity  of 
electricity  which  has  passed.  Thus  the  impulse  of  the  magnet 
is  proportional  to  the  quantity  of  electricity  which  has  passed 
round  the  galvanometer  coils.  Now  in  some  cases  a  finite 
quantity  of  electricity  may  traverse  the  galvanometer  in  a 


280  ELECTRICITY  [CH.  XVII 

very  short  time ;  in  such  a  case  the  current  would  be  very 
large,  but  its  time  of  flow  very  small ;  a  definite  quantity 
of  momentum,  proportional  to  the  total  quantity  of  electricity, 
is  given  to  the  magnet  practically  instantaneously,  and  this 
momentum  will  measure  the  quantity  of  electricity  which  has 
traversed  the  coils. 

Hence  in  such  a  case  the  needle  starts  from  rest  with 
a  momentum  and  therefore  with  an  angular  velocity  which 
is  a  measure  of  the  quantity  of  electricity  which  has  traversed 
the  coils.  Now  the  motion  of  the  needle  resembles  that  of  a 
pendulum  under  gravity ;  it  is  simple  harmonic,  and  in  such 
a  case  the  velocity  of  the  pendulum,  as  it  passes  its  lowest 
point,  is,  when  the  swing  is  small,  proportional  to  the 
amplitude  of  the  swing1. 

Thus  if  a  finite  quantity  of  electricity  traverse  the  coils 
of  a  galvanometer  in  a  very  short  interval  of  time,  the  needle 
receives  an  impulse  which  is  over  before  it  has  moved 
appreciably,  and  in  consequence  it  swings  out  from  its  position 
of  equilibrium,  returning  back  through  that  position  and  finally 
after  some  few  swings  settles  down  to  rest.  The  amplitude 
of  the  first  swing  is  a  measure  of  the  quantity  of  electricity. 
It  can  be  shewn2  that  if  ft  be  the  angular  magnitude  of 
the  first  swing,  T  the  time  of  vibration  of  the  needle,  and 
k  the  reduction  factor  of  the  galvanometer,  then  the  approxi- 
mate relation  between  Q,  the  quantity  of  electricity,  and  ft  is 


T 

=  kT^' 

since  ft  is  very  small. 

Thus  if  we  can  arrange  to  discharge  a  quantity  of  electricity 
through  the  coils  of  a  galvanometer  in  an  interval  of  time 
so  brief  that  we  may  assume  the  magnet  has  not  moved 
appreciably  from  its  equilibrium  position  before  the  flow 
ceases,  we  can  measure  the  quantity  by  observing  the  first 

1  Glazebrook's  Dynamics  §  146. 

2  Glazebrook  and  Shaw's  Practical  Physics,  Chapter  XXI. 


174-175]  MEASUREMENT   OF   QUANTITY  281 

swing  of  the  needle.  If  again  a  second  quantity  is  discharged 
through  the  galvanometer  in  the  same  way  the  ratio  of  the 
two  swings  will  give  us  the  ratio  of  the  two  quantities. 

r\y  Now  when  a  condenser  is  charged  there  are  equal  quantities 
of  positive  and  negative  electricity  respectively  on  its  two 
plates.  If  these  plates  be  connected  through  a  galvanometer 
the  condenser  is  discharged  through  the  galvanometer,  and 
this  discharge  takes  place  in  a  very  brief  interval.  By 
observing  the  swing  of  the  galvanometer  we  can  measure 
the  quantity  passing  in  the  discharge. 

The  galvanometer  which  is  used  for  such  experiments 
should  have  a  long  time  of  swing,  for  it  is  assumed  that 
the  time  of  discharge  is  short  compared  with  the  period  ; 
it  should  also  be  arranged  so  as  not  to  damp  quickly,  for 
the  formula  supposes  that  the  magnet  when  once  disturbed 
will  continue  to  vibrate  like  a  pendulum  for  some  time ; 
if  however  the  experiments  are  merely  relative,  this  last 
requirement  loses  its  importance,  for  the  damping  affects 
all  the  swings  in  the  same  proportion. 

175.  Condensers.  We  proceed  to  describe  some  ex- 
periments on  condensers  in  which  this  ballistic  method  of 
measuring  a  quantity  of  electricity  is  made  use  of. 

In  these  and  for  many  other  purposes  a  Morse  key, 
Fig.  170,  is  useful. 

The  handle  of  the  key 
is  a  lever  carrying  two  con- 
tact pieces  which  make  con- 
tact with  two  studs  on  the 
base  of  the  key.  These  studs 
are  connected  to  two  binding 
screws  A^,  K2.  The  fulcrum  p-  170 

on  which  the  lever  works  is 

attached  to  a  third  binding  screw  K.  A  spring  keeps  the 
handle  raised  so  that  in  the  normal  position  there  is  connexion 
between  K  and  K^.  On  depressing  the  handle  this  connexion 
is  broken  and  contact  is  established  between  K  and  K2. 

EXPERIMENT  46.  To  compare  by  means  of  a  galvanometer 
the  capacities  of  two  condensers. 


282 


ELECTRICITY 


[CH.  XVII 


A  battery  B  of  constant  E.M.F. — e.g.  one  or  two  Daniell's 
cells,  or  preferably  a  storage  cell: — a  condenser  A  and  a 
galvanometer  G  are  connected  through  a  Morse  key  as  shewn 
in  Fig.  171. 


G 


With  the  key  in  the  normal  position  K  and  K^  are  in 
contact  and  the  two  plates  of  the  condenser  are  connected 
through  the  galvanometer;  when  the  handle  is  depressed  Kl 
is  insulated,  while  Kz  is  put  into  connexion  with  I{,  the 
condenser  is  charged  through  the  galvanometer  and  the  first 
throw  of  the  needle  is  proportional  to  the  charge.  Observe 
the  throw  and  wait  until  the  needle  has  again  come  to  rest, 
then  release  the  key.  The  condenser  is  discharged  through 
the  galvanometer  and  the  throw,  in  the  opposite  direction  to 
that  previously  noted,  measures  the  discharge.  If  there  be 
no  leakage  and  no  electric  absorption  the  throws  will  be 
equal. 

Take  a  series  of  readings  on  the  galvanometer  scale  of  the 
throws  due  to  charge  and  discharge,  and  let  the  mean  of 
these  be  bl.  If  the  throw  is  small,  b±  is  approximately 
proportional  to  the  angular  deflexion  /?lf  and  therefore  to 
the  quantity  of  electricity  which  has  passed.  Let  this 
quantity  be  Q1. 

Replace  the  first  condenser  by  the  second  and  repeat  the 
observation.  Let  Qz  be  the  charge  of  the  condenser,  using 
the  same  battery,  and  let  b2  be  the  mean  of  the  throws  as 
measured  on  the  scale. 


175]  MEASUREMENT   OF   QUANTITY  283 

Then  Q2  is  proportional  to  b2. 
Hence  $1  :  Q*  =  &i  •  b2. 

But  if  R  be  the  E.M.F.  of  the  battery,  Clt  C2  the  capacities 
of  the  two  condensers,  we  have 


Thus  Cl  :  C2  =  £>!  :  b.2 

and  the  two  capacities  can  be  compared. 

EXPERIMENT  47.  To  compare  by  the  aid  of  a  condenser 
the  electromotive  forces  of  two  batteries. 

The  measurements  are  the  same  as  in  the  last  experiment 
but  instead  of  changing  the  condenser  we  use  the  same  con- 
denser with  two  different  batteries. 

Let  El,  E2  be  the  electromotive  forces  of  the  batteries, 
C  the  capacity  of  the  condenser,  so  that  now  Ql  is  the  charge 
if  the  condenser  is  charged  to  a  potential  difference  Elt  Q.2  its 
charge  when  the  potential  difference  is  Ez. 

As  before                  Q-^  :  Q2  =  bl  :  62. 
T).-J-  f\   C1  T?      C)  C* T? 

Hence  — -  =  ~ , 

and  thus  the  electromotive  forces  are  compared. 

EXPERIMENT  48.  To  measure  by  means  of  a  ballistic  gal- 
vanometer the  capacity  of  a  condenser. 

The  connexions  are  made  as  in  Fig.  172,  R  is  a  high 
resistance  and  LMNa,  two-way  switch  ;  I{,  K^,  K^  being  a  Morse 
key.  In  the  normal  position  K  and  Kl  are  in  connexion  and 
if  L  and  M  are  connected  the  condenser  is  charged,  the  charge 
however  does  not  traverse  the  galvanometer.  On  depressing 
the  key  the  condenser  is  discharged  through  the  galvanometer 
and  if  Q  be  the  amount  of  the  charge,  /3  the  first  throw  of  the 
needle,  T  the  time  of  swing  and  k  the  reduction  factor, 
then,  as  we  have  seen  in  Section  174, 


284 


ELECTRICITY 


[CH.  XVII 


Now  release  the  Morse  key  and  shift  the  switch  contact 
from  M  to  N.  A  current  from  the  battery  can  flow  through 
the  resistance  R  and  the  galvanometer  of  resistance  G,  the 


Fig.  172. 

amount  of  this  current  is  Ej(B  +  G  +  R}  and  B  the  battery 
resistance  is  usually  negligible  compared  with  R.  Thus  if  6 
be  the  steady  deflexion  of  the  needle  we  have 

E 

^  =  k  tan  0  =  k6, 


since  0  is  small. 
Thus 

Hence 


T 


0' 


and  since  the  quantities  on  the  right  are  known  or  can  be 
observed  the  capacity  C  is  determined. 

In  practice  it  will  be  found  that  unless  R  is  very  large  the  steady 
deflexion  will  be  too  great  to  measure  with  a  galvanometer  sufficiently 
large  to  give  a  measurable  throw  when  the  condenser  is  discharged ; 
it  may  often  be  necessary  to  shunt  the  galvanometer  for  the  second 
part  of  the  experiment  and  in  this  case  the  formula  needs  a  suitable 
modification1. 


1  For  further  details  as  to  these  experiments  reference  should  be  made 
to  Glazebrook  and  Shaw's  Practical  Physics, 


175-176]  MEASUREMENT   OF   QUANTITY  285 

176.  Unit  of  capacity.  The  capacity  of  a  condenser 
is  measured  in  terms  of  a  unit  of  capacity,  and  since  capacity 
is  the  ratio  of  the  charge  to  the  difference  of  potential,  a 
condenser  in  which  unit  charge  produces  unit  potential 
difference  will  have  unit  capacity.  Experiment  shews  that, 
if  we  take  as  the  unit  charge  and  the  unit  potential  difference 
the  electromagnetic  units,  then  the  condenser  of  unit  capacity 
would  be  an  enormously  large  piece  of  apparatus.  It  is  more 
convenient  to  adopt  as  the  unit  the  capacity  of  a  condenser 
which  when  charged  with  1  coulomb  has  a  potential  difference 
between  its  plates  of  1  volt.  This  unit  is  known  as  a  Farad 
(from  Faraday).  Even  this  unit  however  is  far  too  large  for 
ordinary  use. 

DEFINITION.  The  capacity  of  a  condenser  in  which  a 
charge  of  1  coulomb  produces  a  potential  difference  of  1  volt 
is  taken  as  the  practical  Unit  of  Capacity  and  is  called 
i  Farad. 

Since  1  coulomb  is  1/1  Oth  of  the  c.G.s.  unit  of  electricity 
and  1  volt  is  108  c.G.s.  units  of  potential,  a  condenser  whose 
capacity  is  1  farad  contains  1/109  or  10~9  c.G.s.  units  of 
capacity. 

A  farad  however  is  found  to  be  too  big  a  unit  for  most 
purposes  and  as  a  unit  for  ordinary  use  a  Microfarad  or  one- 
millionth  of  a  farad  is  taken. 

A  microfarad  —  10~6  farad  =  10~15  c.G.s.  units  of  capacity. 

If  a  condenser  having  a  capacity  of  1  microfarad  is 
charged  with  1  coulomb  the  potential  difference  between  its 
plates  would  be  one  million  volts. 


CHAPTER  XVIII. 


THERMAL  ACTION  OF  A  CURRENT. 

177.    Heating  of  a    conductor   by   a    current. 

We  have  seen  in  §  113  that  a  conductor  carrying  a  current 
becomes  heated.  This  heat  is  the  equivalent  of  the  energy 
lost  by  the  current  in  passing  along  the  conductor  in  the 
direction  of  the  fall  of  the  potential. 

We  have  also  seen,  §  37,  that  when  a  quantity  of  electricity 
Q  passes  from  a  point  at  potential  Vl  to  a  point  at  lower 
potential  V2  electrical  energy  equal  to  Q  V1  -  Q  V2  disappears. 
In  a  case  in  which  the  electricity  in  its  flow  does  no  external 
work  this  energy,  which  may  be  written  Q  ( Vl  -  F2),  reappears 
as  heat  in  the  conductor. 

If  we  write  E  for  the  fall  of  potential  Vl  -  F2  between  the 
points  where  the  current  enters  and  leaves  the  conductor, 
then  the  loss  of  energy  will  be  EQ,  and  if  the  transfer  be 
due  to  a  current  of  strength  i  flowing  for  t  seconds  Q  =  it. 
Thus  if  W  be  the  work  done,  measured  by  the  loss  of  electrical 
energy,  we  have 

W=EQ  =  E  .i.t. 

Now  let  H  be  the  heat  generated  in  the  conductor  and  J 
Joule's  equivalent1.  Then  since  each  unit  of  heat  is  the 

1  Joule's  equivalent  is  the  amount  of  energy  measured  in  ergs  which, 
if  entirely  transformed  into  heat,  would  raise  1  gramme  of  water  1°  C. 
Glazebrook's  Heat. 


177-178]         THERMAL   ACTION   OF    A    CURRENT  287 

equivalent  of  J  units  of  work,  and  since  an  amount  of  work 
W  is  converted  into  heat  H  we  have  W=JH. 

Thus  JH  =  E  .  i  .  t ; 

or  if  we  write  H'  for  the  rate  at  which  heat  is  produced, 
i.e.  the  amount  of  heat  generated  per  second,  then  H '  =  H/t 
and  hence 


178.  Joule's  Law.  The  above  result  is  known  as 
Joule's  law. 

By  combining  it  with  Ohm's  law  we  can  put  it  in  various 
forms.  For  from  Ohm's  law  E  —  Ri  or  as  we  may  write  it 
i  =  EjR  where  R  is  the  resistance  of  the  conductor  between 
whose  ends  a  difference  of  potential  E  is  maintained. 

E* 

Thus  JH'  =  Ei  =  R?  =  —  . 

K 

Hence  if  a  given  current  flow  through  a  wire  of  given 
resistance,  the  heat  produced  is  proportional  to  the  product 
of  the  resistance  and  the  square  of  the  current,  while  if  a 
given  E.M.F.  is  applied  to  the  ends  of  the  wire  the  heat 
produced  is  proportional  to  the  square  of  the  E.M.F.  divided 
by  the  resistance. 

These  laws  can  be  verified  by  various  experiments. 

EXPERIMENT  49.     To  verify  Joule's  law. 

A  piece  of  thin  silk-covered  wire  of  German-silver  or  some 
other  resistance  material  having  a  resistance  of  4  or  5  ohms 
is  wound  into  a  spiral ;  the  ends  of  the  wire  are  connected  to 
two  thick  leads  of  copper;  the  wire  can  be  immersed  in 
water  in  a  small  copper  calorimeter  (Fig.  173),  the  leads 
passing  through  the  cork  which  closes  the  calorimeter;  a 
stirrer  and  a  thermometer  also  pass  through  the  cork.  The 
coil  forms  part  of  a  circuit  which  contains  a  battery  of  some  5 
or  6  volts  E.M.F.,  an  ammeter  and  a  key. 

The  calorimeter  contains  a  known  mass  M  grammes  of 
water ;  a  resistance  box  may  conveniently  be  included  in  the 
circuit  to  vary  the  current  or  if  preferred  this  can  be  done  by 
altering  the  number  of  cells.  The  temperature  of  the  water 


288 


ELECTRICITY 


[CH.  XVIII 


is  taken,  then  the  circuit  is  closed  and  the  current  allowed  to 
pass  for  10  minutes.  The  strength  of  the  current  is  noted  on 
the  ammeter  and  the  rise  of  temperature  observed,  the  water 


Fig.  173. 

being  well  stirred.  At  the  end  of  the  ten  minutes  the  current 
is  stopped,  the  temperature  continues  to  rise  for  a  short  time 
and  the  highest  point  reached  is  observed.  Thus  the  total  rise 
is  found.  Let  this  be  T°  C.  Then  neglecting  the  loss  by 
radiation  and  the  heat  used  in  raising  the  temperature  of  the 
calorimeter  the  heat  given  out  by  the  wire  has  raised  M 
grammes  of  water  7'°,  its  amount  therefore  is  MT  and  this 
has  been  produced  by  a  current  of  strength  i  amperes  flowing 
for  10  minutes. 

Now  cool1  the  water  down  to  approximately  the  same 
temperature  as  it  had  at  the  beginning,  vary  the  current, 
and  repeat  the  experiment.  Allowing  the  new  current  i'  to 
flow  for  10  minutes,  a  rise  of  temperature  T'°  will  be  ob- 
served ;  and  it  will  be  found  that  we  have  as  approximately 
true  the  relation 


1  By  this  means  the  loss  from  radiation  is  made  to  affect  the  two 
experiments  more  nearly  to  the  same  extent  than  would  be  the  case  if 
the  second  experiment  were  started  at  the  temperature  at  which  the  first 
finishes. 


178]  THERMAL  ACTION   OF  A  CURRENT  289 

The  rise  of  temperature,  since  the  mass  of  water  and  other 
conditions  are  constant,  is  proportional  to  the  heat  caused  by 
the  current,  and  is  found  to  vary  as  the  square  of  the  current. 

EXPERIMENT  50.  To  apply  Joule's  law  to  the  measurement 
of  electromotive  force. 

We  have  from  Joule's  law  the  relation 

JH=Eit, 
where  H  is  the  heat  produced  in  t  seconds. 

Hence  E  =  ,  —  . 

i  .  t 

The  previous  experiment  has  enabled  us  to  measure  the 
various  quantities  on  the  right-hand  side  with  the  exception 
of  J.  We  have  seen  (Glazebrook,  Heat,  §  176)  how  this 
constant  can  be  measured,  and  its  value  has  been  shewn  to 
be  42  x  106  ergs.  The  value1  of  H  is  MT  where  M  is  the 
mass  and  T  the  rise  of  temperature  of  the  water.  The 
current  i  should  be  measured  in  C.G.S.  units,  since  the  other 
quantities  are  in  C.G.S.  units,  hence  if  the  ammeter  reads 
directly  in  amperes  and  if  A  be  the  reading  observed  i  =  A/W. 


4-2  x          ^f 

Hence  E  =       -  ----  108; 

jfLt 

t  is  the  time  of  flow  of  the  current  which  we  have  taken  to  be 
10  minutes. 

The  value  of  E  is  given  in  C.G.S.  units  of  electromotive 
force.  Since  1  volt=108  C.G.S.  units,  in  order  to  find  the 
value  in  volts  we  must  divide  by  10®  and  we  thus  get 

4-2  x  MT 


At 


volts. 


Example.  A  current  of  -75  ampere  flows  through  the  wire  for  10 
minutes.  The  mass  of  water  in  the  calorimeter  is  85  grammes  and  the 
rise  of  temperature  5°-4  C.  Find  the  potential  difference  betiveen  the 
ends  of  the  wire. 

1  This  value  requires  correcting  (1)  for  the  loss  by  radiation,  (2)  for 
the  heat  given  to  the  calorimeter  and  stirrer  :  for  the  method  of  applying 
these  corrections,  see  Glazebrook's  Heat  and  Glazebrook  and  Shaw's 
Practical  Physics. 

G.  E.  19 


290  ELECTRICITY  [CH.  XVIII 

We  have  M=85,  T=5°-4,  A--15,  t  =  600  seconds. 

,     4-2x85x5-4 

Hence  E=—  -  =4-28  volts. 

450 

Again  from  the  above  we  can  calculate  the  resistance  of  the  wire 
in  which  the  current  is  flowing,  for  the  current  of  -75  ampere  is  pro- 
duced by  an  E.M.F.  of  4-28  volts. 

Hence  the  resistance  is  4-2S/-75  or  5-71  ohms. 

179.  Joule's  law  applied  to  a  battery.  It  should 
be  noticed  that  in  the  above  sections  E  stands  for  the  difference 
of  potential  between  the  ends  of  the  wire  and  R  the  resistance 
of  the  wire. 

We  can  however  apply  the  same  reasoning  to  the  complete 
circuit  including  the  battery.  If  E  be  the  electromotive  force 
of  the  battery  a  quantity  of  electrical  energy  EQ  disappears 
when  Q  units  of  electricity  pass  completely  round  the  circuit, 
and  if  no  external  work  is  done  this  reappears  as  heat  in  the 
various  conductors  of  which  the  circuit  is  composed.  If  we 
write  Q  =  it  arid  E  =  (B  +  R)i,  B  being  the  battery  resistance, 
R  that  of  the  rest  of  the  circuit,  then 


and  of  this  heat  an  amount  Bi^tfJ  appears  in  the  battery, 
RtftjJ  in  the  external  part  of  the  circuit. 

Example.  A  battery  whose  resistance  is  half  an  ohm  is  producing 
a  current  of  15  amperes  in  a  circuit  whose  resistance  is  10  ohms,  find 
the  heat  generated  per  second  in  the  battery  and  in  the  circuit.  Find 
also  the  E.M.F.  of  the  battery. 

We  have  E  =  Ri  =  l5  x  10-5  =  157 '5  volts. 

In  calculating  the  heat  we  must  remember  that  in  this  formula  both 
i  and  R  are  to  be  measured  in  C.G.S.  units. 

Hence  i  =  15  amperes  =  1 -5  C.G.S.  units, 

R  =  1Q  ohms  =  10  x  109  C.G.S.  units, 
B=  '5  ohm  =  -5  x  109  C.G.S.  units, 
J=4-2x  107  ergs. 
Hence  for  the  battery 

1-5  x  1-5  x  -5  x  109     , 


4-2  x 
and  for  the  wire 


Heat=1'5x^°X11°xl0'  =  S85-6  units. 
4  ^  x  J-0 


178-181]          THERMAL   ACTION   OF   A   CURRENT  291 

The  total  heat  is  given  by  the  formula 


and  since  we  have  seen  that  jE  =  157'5  volts,  i  =  lo  amperes,  we  obtain 

Heat  =  562  -5  units 
and  this  is  very  nearly1  the  sum  of  26*8  and  535-6. 

180.  Electrical  Energy.     Whenever  a  quantity  Q 
of  electricity  is  transferred  round  a  circuit  in  which  there 
is  an  E.M.F.  E,  the  work  done  as  measured  by  the  electrical 
energy  is  KQ  ;  if  E  and  Q  are  both  measured  in  c.G.s.  units 
this  electrical  energy  is  measured  in  ergs;    it  is  more    con- 
venient however  to  measure  the  E.M.F.  in  volts,  the  current 
in  amperes  and  the  quantity  transferred  in  coulombs,  to  adopt, 
that  is,  the  practical  rather  than  the  c.G.s.  system  of  units. 

The  unit  of  energy  011  this  system  will  be  the  work  done 
when  1  coulomb  is  conveyed  round  a  circuit  under  an  E.M.F. 
of  1  volt.  This  unit  is  called  a  Joule. 

DEFINITION.  A  Joule  is  the  amount  of  electrical  energy 
expended  by  the  transference  of  1  coulomb  round  a  circuit  in 
which  the  electromotive  force  is  1  volt. 

Since  1  coulomb  =  ^  c.G.s.  unit  and  1  volt  =  108  C.G.S. 
units,  we  see  that 

1  joule  =  107  C.G.S.  units  of  work  =  107  ergs. 

Moreover  since  Joule's  equivalent  has  been  shewn  to  be 
4-2  x  107  ergs,  we  see  that 

Joule's  equivalent  —  4-2  joules. 

181.  Electrical    Power.     Power   has   been   defined2 
as  the  rate  of  doing  work,  it  is  measured  therefore  by  the 
number  of  units  of  work  done  per  second.     In  c.G.s.  units  it  is 
the  number  of  ergs  done  per  second. 

It  is  convenient  to  have  a  practical  unit  of  power  related 
to  the  joule.  This  is  called  the  Watt  and  is  the  power 
expended  when  work  is  being  done  at  the  rate  of  1  joule 
per  second. 

1  The  difference  is  due  to  not  carrying  the  figures  in  the  arithmetic  to 
a  sufficient  number  of  decimal  places. 

2  Glazebrook,  Dynamics,  §  110. 

19—2 


292  ELECTRICITY  [CH.  XVIII 

One  joule  is  done  if  a  coulomb  is  transferred  round  the 
circuit  under  an  E.M.F.  of  L  volt.  Now  if  a  coulomb  is 
transferred  per  second  the  current  is  1  ampere,  hence  if  a 
current  of  1  ampere  is  flowing  under  an  E.M.F.  of  •!  volt  the 
power  expended  is  1  watt. 

DEFINITION.  A  power  of  1  Watt  is  expended  in  pro- 
ducing a  current  of  1  ampere  under  an  electromotive  force 
of  1  volt. 

Moreover  since  1  ampere  =  1/10  c.G.s.  unit  of  current  and 
1  volt  =  108  c.G.s.  units  of  E.M.F. 

1  watt  =  107  c.G.s.  units  of  power  =  107  ergs  per  second. 

Thus  in  any  given  circuit  the  power  in  watts  is  found  by 
multiplying  the  volts  by  the  amperes,  while  the  work  done  in 
a  given  time  is  found  in  joules  by  multiplying  the  power  in 
watts  by  the  time  in  seconds. 

A  unit  of  power  often  employed  in  practice  is  the  kilowatt, 
or  1000  watts.  This  is  known  as  the  Board  of  Trade  unit. 

The  relation  between  the  erg  and  the  foot-pound  is  known, 
and  it  can  be  shewn  that1 

1  erg  =  -737  x  10~7  foot-pound. 
Thus  it  follows  that 

1  joule  =  -737  foot-pound, 
and  1  foot-pound  =  1-356  joules. 

Again  when  work  is  done  at  the  rate  of  550  foot-pounds 
per  second  1  horse-power  is  exerted. 

Hence 

1  horse-power  =  550  x  1-356  joules  per  second 

-  746  watts. 

1  watt  =  1/746  =  -001340  horse-power. 
1  kilowatt—  1'34  horse-power. 

EXPERIMENT  51.  To  determine  Joule's  equivalent  by  elec- 
trical measurements. 

The  experiments  described  in  Section  177  can  be  utilized 
to  find  Joule's  equivalent.  For  we  have  the  equation  JH  -  Eit. 

1  Glazebrook,  Dynamics,  §  110. 


181-182]          THERMAL   ACTION    OF   A   CURRENT  293 

Now  E  and  i  can  be  found  by  electrical  measurements,  and 
If  is  determined  as  in  Experiment  50,  by  the  calorimeter,  and 
then  from  the  equation  J=  EitjH  we  find  J. 

182.  Thermo-electricity.  If  a  circuit  be  composed 
of  two  different  metals,  and  if  one  of  the  two  junctions  of  the 
metals  be  at  a  different  temperature  to  the  other,  a  current  is 
produced  round  the  circuit. 

This  current  which  may  be  shewn  in  various  ways  is  said 
to  be  due  to  Thermo-electric  action. 

EXPERIMENT  52.  To  shew  the  thermo-electric  production  of 
a  current. 

(a)  A  strip  of  copper  and  one  of  German-silver  are  brazed 
together  at  their  ends  and  bent  so  as  to  form  a  narrow 
rectangle,  as  shewn  in  Fig.  174.  This  is  placed  with  its  long 
edges  horizontal  and  a  compass- needle  is  mounted  so  that  its 
centre  may  coincide  with  that  of  the  rectangle. 


The  plane  of  the  rectangle  lies  in  the  meridian,  parallel 
to  the  axis  of  the  compass-needle.  On  heating  one  of  the 
junctions  with  a  Bunsen-burner  it  will  be  noticed  that  the 
compass-needle  is  deflected  ;  a  current  is  traversing  the  circuit. 
Note  the  direction  of  the  deflexion  and  infer  thence  the 
direction  of  the  current. 

(b)  Two  pieces  of  copper-wire  are  fastened  to  the  ends  of 
an  iron  rod.  This  may  be  done  by  drilling  a  hole  through 
the  rod,  passing  the  copper-wire  through  the  hole  and  winding 


294  ELECTRICITY  [CH.  XVIII 

it  tightly  for  a  few  turns  round  the  rod.  The  other  ends  of 
the  wires  are  connected  to  an  ammeter  or  low  resistance 
galvanometer;  one  of  the  junctions  is  heated  by  a  Bunsen- 
burner  and  the  galvanometer  needle  is  deflected.  Notice  the 
direction  of  the  deflexion  and  so  find  that  of  the  current. 

183.  Observations  on  Thermo-electricity.  It  will 
be  found  that  if  in  the  last  experiment  one  junction  be  kept 
at  the  ordinary  temperature  while  the  other  is  gradually 
heated,  the  current  will  at  first  pass  from  copper  to  iron 
through  the  hot  junction,  and  this  current  will  rise  as  the 
temperature  of  the  hot  junction  is  raised  until  that  tem- 
perature reaches  a  limiting  value  T,  known  as  the  neutral 
temperature  for  these  two  metals.  In  the  case  of  iron  and 
copper  this  neutral  temperature  is  about  284°  C.  When  the 
temperature  of  the  hot  junction  is  raised  above  the  neutral  tem- 
perature the  E.M.F.  in  the  circuit  arid  in  consequence  the  current 
decrease,  and  this  continues  until  the  hot  junction  is  as  much 
above  the  neutral  point  as  the  cold  junction  is  below,  when 
the  current  is  again  zero ;  if  the  hot  junction  be  still  further 
raised  in  temperature  the  direction  of  the  current  is  reversed. 

The  thermo-electric  force  produced  by  a  given  difference 
of  temperature  varies  very  much  for  different  metals,  and 
it  is  possible  to  arrange  the  metals  in  a  list  so  that  if  two  of 
them  are  joined  together  the  current  will  pass  from  the  hot 
junction  to  the  cold  in  that  metal  of  the  two  which  stands 
last  on  the  list.  In  this  statement  it  is  assumed  that  the 
mean  temperature  of  the  two  junctions  is  below  their  neutral 
temperature. 

Thus  at  ordinary  temperatures  bismuth,  German-silver, 
lead,  platinum,  copper,  zinc,  iron,  antimony  form  such  a 
list.  In  thermo-electric  experiments  it  is  usual  for  a  reason 
which  will  appear  later  to  take  lead  as  the  standard  metal 
and  to  refer  other  metals  to  it.  As  the  temperature  of  the 
hot  junction  rises  the  electromotive  force  changes,  and  the 
rate  of  increase  of  the  E.M.F.  for  each  rise  of  temperature  of  1°C. 
at  any  given  temperature  is  called  the  thermo-electric  power 
at  that  temperature.  In  the  case  of  metals  which  stand  before 
lead  on  the  list  the  thermo-electric  power  is  positive ;  the 
thermo-electric  force  is  from  the  hot  junction  to  the  cold  and 


182-183]          THERMAL   ACTION    OF   A    CURRENT 


295 


is  increased  by  a  rise  of  temperature ;  in  the  case  of  metals 
below  lead  the  thermo-electromotive  power  is  negative ;  the 
thermo-electromotive  force  from  the  hot  junction  to  the  cold 
increases  negatively  with  rise  of  temperature,  that  is,  the 
electromotive  force  is  as  in  the  case  of  the •  copper-iron  junction 
from  the  cold  junction  to  the  hot. 

By  measuring  the  E.M.F.  in  a  given  circuit  in  which  one 
junction  is  kept  at  some  convenient  temperature,  say  0°  C., 
while  the  other  is  heated,  and  plotting  the  results  we  can 
obtain  a  curve  in  which  the  ordinates  give  the  thermo-electro- 
motive force  and  the  abscissae  the  temperature,  and  from 
these  curves  the  thermo-electromotive  power  which  is  the  rate 
at  which  the  E.M.F.  increases  with  the  temperature  can  be 
calculated.  Thus  a  second  series  of  curves  of  thermo-electro- 
motive power  can  be  plotted. 


15 


5O    10O    150   2OO   250   300   350   400   450   50O   550   600 


15 


5O    1OO    ISO   200   250   30O   35O   4OO   450   5OO   550   60O 


Fig.  175. 

For  a  large  number  of  metals  and  over  a  considerable 
range  of  temperature  these  curves  of  thermo-electric  power 
are  found  to  be  straight  lines.  The  same  experiments  shew 
that  the  curves  of  thermo-electric  force  are  parabolas. 

Fig.    175    gives   a    thermo-electric   diagram    or   series    of 


296  ELECTRICITY  [CH.  XVIII 

curves  of  thermo-electric  power,  the  abscissae  represent 
temperatures,  the  ordinates  give  the  thermo-electric  power 
measured  in  microvolts  per  degree  of  temperature. 

The  thermo-electric  power  at  any  temperature  will  measure 
very  approximately  the  E.M.F.  round  a  circuit  of  the  given 
metal  and  lead  when  one  junction  is  J  a  degree  above,  the 
other  \  a  degree  below,  the  given  temperature.  When  the 
thermo-electric  power  is  positive  it  means  that  the  current 
passes  from  the  hot  junction  to  the  cold  jn_the  lead.  Thus 
bismuth  is  therm o-electrically  positive  with  regard  to  lead 
since  the  current  passes  at  the  hot  junction  from  the  bismuth 
to  the  lead.  Antimony  again  is  thermo-electrically  negative, 
for  at  the  hot  junction  the  current  is  from  lead  to  antimony. 

The  electromotive  forces  caused  by  heating  a  single  junction 
are  very  small ;  thus  if  one  junction  of  a  copper-iron  circuit 
be  at  1°  C.,  the  other  being  at  0°  C.,  the  E.M.F.  will  be  about 
15  microvolts1 ;  in  the  case  of  a  bismuth-antimony  circuit  it 
is  greater,  being  about  110  microvolts;  some  other  metals 
and  alloys  have  a  still  greater  value  than  this,  but  in  all  cases 
the  thermo-electromotive  force  is  small  compared  with  those 
which  arise  from  chemical  action. 

If  a  thermo-electric  diagram  be  drawn  it  will  be  noticed 
that  the  lines  for  the  different  metals  intersect  each  other, 
the  temperature  corresponding  to  the  intersection  of  the  lines 
of  two  metals  gives  the  neutral  temperature  for  those  two 
metals.  At  the  neutral  point  the  two  metals  have  the  same 
thermo-electric  power. 

184.  Peltier  effect.  The  discovery  that  there  is  an 
electromotive  force  in  an  unequally  heated  circuit  of  two 
metals  is  due  to  Seebeck.  The  converse  of  this  fact,  viz.  that 
if  a  current  be  made  to  flow  across  the  junction  it  heats  it  if 
it  flows  in  one  direction,  cools  it  if  it  flows  in  the  other,  is  due 
to  Peltier.  Seebeck  shewed  that  in  a  copper-iron  circuit  a 
current  will  pass  from  copper  to  iron  across  the  heated 
junction.  Peltier  proved  that  if  a  current  be  made  to  pass 
from  copper  to  iron  across  a  junction  that  junction  is  cooled, 
if  however  the  current  flows  from  iron  to  copper  the  junction 

1  1  microvolt^ one  millionth  of  a  volt  =  10~6  volt. 


183-184]          THERMAL   ACTION   OF   A    CURRENT 


297 


is  heated.  This  heating  effect  must  be  distinguished  from  the 
Joule  effect ;  the  resistance  of  the  circuit  causes  the  evolution 
of  heat  whichever  way  the  current  flows ;  the  Peltier  effect  is 
reversible  with  the  current ;  in  one  direction  heat  is  evolved 
with  the  current,  in  the  other  it  is  absorbed  and  the  junction 
is  cooled. 

The  Peltier  effect  is  more  marked  in  two  metals  such  as 
bismuth  and  antimony  which  are  at  some  distance  apart  in 
the  thermo-electric  series  than  between  two  such  as  copper 
and  iron. 

When  a  bismuth-antimony  junction  is  heated  the  current 
flows  from  bismuth  to  antimony ;  if  a  current  be  passed  across 
the  junction  in  this  direction  the  junction  is  cooled.  This  can 
be  proved  by  enclosing  one  of  the  two  junctions  in  the  bulb  of 
an  air  thermometer  as  in  Fig.  176.  The  two  metals  pass 
through  tubulures  at  either  side 
of  the  bulb  and  are  sealed  into 
it  with  some  air-tight  cement. 
On  passing  a  current  from  anti- 
mony to  bismuth  the  junction 
is  heated  and  the  column  of 
liquid  in  the  thermometer  tube 
falls ;  on  reversing  the  current 
the  junction  is  cooled  and  the 
column  rises.  It  must  be  noted 
however  that  the  metal  is  being 
heated  in  consequence  of  its  re- 
sistance ;  this  heats  the  air  in 
the  bulb,  and  unless  this  heating 
be  less  than  the  cooling  due  to 
the  Peltier  effect  the  column 
will  not  rise  in  the  latter  case, 
but  will  fall  more  slowly  than 
in  the  former.  This  difficulty 
may  be  overcome  by  the  use  of  Fig.  176. 

a   differential    air    thermometer 

as  shewn  in  Fig.  177,  the  one  junction  is  in  one  bulb,  the 
other  in  the  second  bulb.  The  Joule  heating  effect  is  the 
same  for  the  two  and  the  motion  of  the  liquid  will  depend 


298 


ELECTRICITY 


[CH.  XVIII 


on  the   difference  of  the  heating  effect  in  one  bulb  and  the 
cooling  effect  in  the  other. 


war       t        pffif  wtir       [        itrf 


Fig.  177. 

The  heat  produced  at  the  junction  is  found  to  be  pro- 
portional to  the  current  passing  and  to  the  time,  i.e.  to  the 
quantity  of  electricity  which  has  crossed  the  section.  Thus  if 
i  be  the  current  in  c.G.s.  units  and  t  the  time  in  seconds  the 
energy  absorbed  or  evolved  at  the  junction  may  be  written 
P .i.t,  and  P  is  a  coefficient  which  is  called  the  coefficient  of 
the  Peltier  effect  and  is  measured  by  the  energy  evolved  by 
the  passage  of  1  c.G.s.  unit  of  electricity. 

Moreover  it  can  be  shewn  by  aid  of  the  thermo-electric 
diagram  that  the  thermo-electric  power  at  any  junction  is 
found  by  dividing  the  coefficient  of  the  Peltier  effect  by  the 
temperature  of  the  junction  measured  from  absolute  zero. 

185.  Thomson  effect.  It  was  shewn  by  Lord  Kelvin 
that  the  passage  of  a  current  along  an  unequally  heated  con- 
ductor of  any  material  except  lead  causes  the  absorption  or 
evolution  of  heat  at  each  point  of  the  conductor  according  to 
the  direction  of  the  current.  In  the  case  of  copper  heat  is 
absorbed  and  the  wire  cooled  if  the  current  flows  from  the 
cold  part  of  the  wire  to  the  hot  part ;  in  the  case  of  iron  the 


184-1 86 J          THERMAL   ACTION    OF    A    CURRENT 


299 


reverse  is  true ;  this  is  known  as  the  Thomson  effect.  If  the 
direction  of  the  current  he  reversed  the  absorption  of  heat 
becomes  an  evolution  and  conversely. 

The  absence  of  the  Thomson  effect  in  lead  is  the  reason 
why  lead  is  chosen  as  the  standard  metal  in  thermo-electric 
measurements. 

186.  The  Thermopile.  The  thermo-electromotive 
force  of  an  antimony  and  bismuth  couple  is  made  use  of  in 
a  thermopile  for  the  measurement  of  small  differences  of 
temperature.  A  number  of  bars  of  these  metals  are  arranged 
alternately,  as  in  Fig.  178,  A  1,  2  3,  4  5,  being  antimony  bars 
and  12,  3  4,  5  B  bismuth  A 

bars ;  the  bars  are  soldered 
together  at  1,  2,  3,  4,  5,  the 
ends  A  and  B  being  con- 
nected to  a  low  resistance 
galvanometer.  If  the  junc- 
tions 1,  3,  5  be  heated  while 
2  and  4  remain  cool  a  Fi  17g 

current    will    flow    through 

the  galvanometer  from  A  to  B.  If  the  junctions  2,  4  be 
heated,  1,  3,  5  remaining  cool,  the  direction  of  the  current 
will  be  reversed.  The  E.M.F.  is  proportional 
to  the  number  of  junctions,  hence  in  the 
apparatus  as  usually  made  a  large  number 
of  junctions  are  connected  up  in  square 
order,  as  shewn  in  Fig.  179,  the  con- 
tiguous bars  of  metal  being  insulated  from 
each  other  electrically  by  strips  of  mica. 

The  same  principle  is  applied  to  the 
measurement  of  high  temperatures ;  a 
couple  made  of  wires  of  pure  platinum 
and  an  alloy  of  platinum-rhodium  or  platinum-indium  is 
employed.  By  means  of  preliminary  experiments,  either  by 
a  comparison  with  a  standard  thermo-couple  or  by  the  aid 
of  a  suitable  air  thermometer,  a  curve  giving  the  relation 
between  E.M.F.  and  temperature  is  plotted  for  the  couple. 
After  this  has  been  done  a  determination  of  the  E.M.F.  under 
any  conditions  will  give  the  temperature  of  the  junction. 


Fig.  179. 


300  ELECTRICITY  [CH.  XVIII 

187.  Platinum  Thermometer.  Another  electrical 
method  of  measuring  temperature  depends  on  the  fact  that 
the  resistance  of  a  wire  rises  as  its  temperature  is  increased. 
If  then  we  know  (1)  the  resistance  of  a  piece  of  wire  at  some 
given  temperature  0°  C.  (say),  and  (2)  the  rate  at  which  the 
resistance  increases  with  the  temperature,  then  we  can  by 
measuring  the  resistance  of  the  wire  determine  its  temperature. 

For  if  R0  be  the  resistance  at  0°  C.  and  aR0  the  change  in 
resistance  for  each  degree  centigrade ;  then  assuming  for  the 
present  that  a  is  a  constant  the  increase  of  resistance  for 
t°  C.  is  aR0t.  Hence  if  R  be  the  resistance  at  t°  we  have 

and  therefore 

To  find  a  we  have  if  /?100  be  the  resistance  at  100°  C. 
Hence  a  = 


?0  +  a/?0.100. 

7?inn  —  Rn 


ioo^0  • 

Thus  1  =  100^4. 

**1W  ~  ^0 

The  wire  usually  employed  for  the  purpose  is 'platinum 
which  is  carefully  annealed ;  now  it  has  been  shewn  by 
Professor  Calleiidar  that  the  coefficient  of  increase  of  resistance 
of  platinum  per  degree  centigrade  is  not  the  same  for  all 
temperatures,  so  that  if  the  symbol  t  in  the  above  formulae  is 
taken  to  mean  temperature  in  degrees  centigrade  they  are 
not  exactly  true.  We  may  however  adopt  a  scale  of  tempera- 
ture for  which  they  are  exact ;  this  is  known  as  the  platinum 
scale  and  a  rise  of  1°  on  the  platinum  scale  is  a  rise  of 
temperature  which  produces  a  change  in  the  resistance  of  the 
wire  of  one  hundredth  the  amount  occurring  between  the 
freezing  point  and  the  boiling  point. 

If  we  agree  to  reckon  temperatures  by  this  platinum  scale 
then  the  equation 

R-RQ 


t    -100 

l'pt  —  L  uu 


is  true. 


187]        THERMAL  ACTION  OF  A  CURRENT       301 

Moreover  Professor  Callendar  has  shewn  that  there  is  a 
very  simple  connexion  between  temperature  measured  on  the 
platinum  scale  and  temperature  measured  on  the 
centigrade  scale,  which  is  given  by  the  equation 


where  8  is  a  constant,  depending  on  the  wire, 
which  for  pure  platinum  differs  very  little  from 
the  value  1-5. 

The  wire  employed  as  a  thermometer  is  coiled 
on  a  mica  frame  and  enclosed  inside  a  glass  or 
porcelain  tube  which  can  be  immersed  in  the 
material  whose  temperature  is  to  be  measured 
in  the  same  way  as  an  ordinary  thermometer; 
the  resistance  of  the  wire  is  then  measured ;  this 
gives  us  R,  and  hence  _S0  and  /?100  being  known 
from  preliminary  observations  we  can  find  tpt ; 
then  t  is  given  by  the  formula.  With  the  usual 
value  of  8  the  difference  between  t  and  tpt  at 
50°  C.  is  0°'37  while  at  500°  C.  it  is  about  30°. 

There  is  an  arrangement  whereby  compensa- 
tion is  secured  for  the  resistance  of  the  connexions 
leading  to  the  coil  so  that  the  resistance  of  the       Fig.  180. 
platinum  spiral  alone  is  measured. 

Fig.   180  shews  a  platinum  thermometer  as  usually  con- 
structed. 


CHAPTEE   XIX. 


THE  VOLTAIC   CELL.     (THEORY.) 

188.  Energy  changes  in  a  cell.  We  have  already 
seen,  §  115,  that  if  two  platinum  plates  are  immersed  in 
acidulated  water  and  a  potential  difference  maintained  between 
them,  then  positive  electricity  flows  from  the  plate  at  higher 
potential  to  that  at  lower,  while  oxygen  is  deposited  at  the 
first  plate,  hydrogen  at  the  second,  and  moreover  that  for 
each  gramme  of  hydrogen  deposited  on  the  latter,  8  grammes 
of  oxygen  are  deposited  on  the  former.  Again  if  two  metallic 
plates  of  different  materials  are  placed  in  dilute  sulphuric 
acid,  let  us  say  copper  and  zinc,  and  connected  by  a  copper 
wire  outside  the  acid,  then  positive  electricity  passes  through 
the  acid  from  the  zinc ;  hydrogen  collects  on  the  copper  and 
for  each  gramme  of  hydrogen  deposited,  32-5  grammes  of 
zinc  are  taken  from  the  zinc  plate  winch  combine  with 
48  grammes  of  sulphion,  SO4,  to  form  80*5  grammes  of  zinc 
sulphate. 

Now  it  is  from  this  combination  of  the  zinc  and  acid 
that  the  energy  required  to  drive  the  current  is  produced 
and  we  can  obtain  a  relation  between  the  electromotive  force 
of  the  battery  and  the  energy  of  chemical  combination  thus. 

Let  us  call  H  the  energy  set  free  when  one  gramme  of 
zinc  combines  with  oxygen ;  it  is  known  as  the  heat1  of 

1  We  here  suppose  the  heat  to  be  measured  as  energy  :  if  we  take 
h  as  the  heat  of  combination  in  calories  and  J  as  Joule's  equivalent,  we 
have  H=Jh. 


188-189]  THE   VOLTAIC   CELL  303 

combination  of  zinc  and  oxygen,  and  can  be  determined  from 
caloriinetric  observations.  Then  if  a  quantity  Q  of  positive 
electricity  pass  from  the  zinc  to  the  acid,  and  if  y  be  the 
electrochemical  equivalent  of  zinc,  a  mass  of  zinc  Qy  grammes 
is  removed  and  sets  free  an  amount  of  energy  HQy. 

The  result  is  the  transference  of  a  quantity  Q  round  the 
circuit,  and  if  E  be  the  E.M.F.  of  the  battery  the.  work  needed 
to  do  this  is  EQ  units  of  work. 

If  we  assume  that  this  work  is  derived  from  the  chemical 
combination,  then  we  must  have 


or  E  =  Hy. 

We  thus  express  the  E.M.F.  of  the  cell  in  terms  of  the 
heat  of  combination  of  the  zinc  and  oxygen  and  the  electro- 
chemical equivalent  of  the  metal. 

This  simple  theory  requires,  as  Helmholtz  shewed,  some  modification 
from  the  fact  that  the  E.M.F.  of  a  cell  depends  on  its  temperature,  while 
in  consequence  of  the  passage  of  the  current  reversible  thermal  changes 
go  on  at  various  junctions,  but  in  most  cases  in  practice  the  correction 
introduced  by  this  consideration  is  small. 

189.     Electromotive  Force  of  a  DanielFs  Cell. 

In  the  above  we  have  assumed  that  all  the  chemical  inter- 
changes occur  at  the  zinc  plate;  the  whole  energy  of  the 
cell  comes  in  this  case  from  this  change.  In  reality  in  most 
cells  changes  occur  elsewhere  which  give  rise  to  the  absorption 
or  liberation  of  energy,  and  when  all  these  changes  are  taken 
into  consideration  the  calculated  E.M.F.  is  found  to  agree 
closely  with  that  observed. 

Thus  if  we  suppose  that  in  a  Daniell's  cell  energy  is 
liberated  by  the  removal  of  the  zinc  and  absorbed  by  the 
deposition  of  the  copper,  and  that  no  other  chemical  actions 
involving  liberation  or  absorption  of  energy  occur,  we  proceed 
to  calculate  the  E.M.F.  as  follows. 

It  lias  been  shewn  that  the  combination  of  32-5  grammes 
of  zinc  to  form  zinc  sulphate  liberates  54231  calories,  while 
the  deposition  of  the  corresponding  amount  31'6  grammes.  of 
copper  from  copper  sulphate  absorbs  27112  calories,  thus 


304  ELECTRICITY  [CH.  XIX 

the  heat  equivalent  of  the  energy  liberated  on  the  passage  of 
the  quantity  of  electricity  required  to  dissolve  this  quantity  of 
zinc  is  the  difference,  or  27119  calories. 

This  quantity  of  electricity  deposits  1  gramme  of  hydrogen, 
and  since  the  C.G.S.  unit  of  electricity  deposits  0-0001038 
grammes  of  hydrogen,  the  quantity  of  electricity  required  to 
deposit  1  gramme  is  1/-0001038  or  9634  C.G.S.  units. 

Thus  the  quantity  of  electricity  which  has  passed  is 
9634  C.G.S.  units. 

Now  since  the  value  of  the  mechanical  equivalent  of  heat 
is  4*2  x  107  ergs,  the  energy  liberated  is 

27119  x  4-2  x  107ergs, 

and  the  E.M.F.  produced  being  the  energy  liberated  on  the 
passage  of  unit  quantity  of  electricity  is 

27119  x  4-2  x  107/9634  C.G.S.  units. 

This  it  will  be  found  reduces  to  1  -18  x  108  C.G.S.  units,  or  since 
1  volt  is  108  units  we  have  for  the  E.M.F.  of  the  cell  the  value 
1-18  volts.  We  have  thus  calculated  the  E.M.F.  of  the  cell 
from  the  chemical  changes. 

The  result  is  some  5  or  6  per  cent,  greater  than  the  value  found 
by  direct  experiment,  but  the  latter  depends  on  the  concentration  of  the 
zinc  sulphate  and  in  the  theoretical  account  various  small  corrections 
have  been  omitted. 


19O.  Chemical  and  Electrical  Transformations 
in  Electrolysis.  Let  us  now  endeavour  to  picture  to 
ourselves  what  is  going  on  when  acidulated  water  is  being 
decomposed  in  an  electrolytic  cell,  or,  when  a  voltaic  cell  is 
producing  a  current,  and  see  if  we  can  arrive  at  any 
satisfactory  theory  of  the  action  of  the  cell. 

A  molecule  of  water  consists  of  two  atoms  of  hydrogen 
and  one  of  oxygen.  With  the  water  are  molecules  of  sulphuric 
acid  H2SO4.  Under  the  action  of  the  electric  forces  these  are 
decomposed  ;  the  hydrogen  atoms  carrying  positive  electricity 
move  with  the  positive  current  to  the  negative  plate.  Sul- 
phion,  SO4,  carrying  a  negative  charge  is  set  free,  this 


189-190]  THE   VOLTAIC   CELL  305 

re-combines  with  two  atoms  of  hydrogen  from  the  water 
to  form  sulphuric  acid  H2S04  setting  free  an  atom  of  oxygen 
with  its  corresponding  negative  charge  to  appear  at  the 
positive  plate. 

Again  since  in  the  electrolysis  of  water  the  masses  of 
oxygen  and  hydrogen  deposited  in  a  given  time  are  in  the 
ratio  of  8  to  1,  while  the  masses  of  the  respective  atoms 
are  as  16  to  1,  it  follows  that  the  number  of  hydrogen  atoms 
deposited  in  that  time  is  twice  as  great  as  that  of  oxygen 
atoms :  but  since  the  quantities  of  positive  and  negative 
electricity  concerned  are  equal  the  charge  of  each  oxygen 
atom,  if  we  suppose  the  electricity  carried  by  the  atoms, 
is  twice  as  great  as  that  of  an  hydrogen  atom ;  this  is  found 
to  be  the  case  for  any  divalent  constituent  of  a  salt. 

Hence  if  we  call  e,  the  charge  of  an  atom  of  hydrogen,  the 
charge  of  the  atom  of  oxygen  is  -  '2e,  that  of  the  group  SO4  is 
also  -  2e,  while  since  in  copper  sulphate  or  in  zinc  sulphate 
the  atom  of  copper  or  zinc  replaces  respectively  two  atoms 
of  hydrogen  the  charge  is  in  each  case  +  2e.  In  the  case 
of  silver  nitrate  however,  which  is  univalent,  the  charge  on 
each  atom  of  the  silver  is  +  e. 

Thus  we  are  to  look  upon  sulphuric  acid  as  a  series  of 
molecules  H2SO4  in  which  each  of  the  hydrogen  atoms  carries 
a  charge  -t-  e,  while  the  SO4  group  carries  —  2e ;  the  whole 
charge  therefore  is  zero,  being  made  up  of  +  '2e  on  the  two 
hydrogen  atoms  and  —  2e  on  the  SO4. 

It  may  possibly  be  better  to  consider  the  group  S04  as  S03  with  a 
charge  -e  and  0  also  with  a  charge  -e. 

When  electric  force  acts  on  the  liquid  some  of  the  mole- 
cules are  apparently  split  up  into  H.,  with  +  2e  and  SO4  with 
-2«. 

There  are  however  a  number  of  facts  which  point  to  the 
conclusion  that  the  union  between  two  or  more  atoms  to  form 
a  molecule  is  not  a  permanent  one,  but  that  continuous  inter- 
changes of  partners  are  always  going  on  among  the  molecules. 
At  any  moment  by  far  the  greater  number  of  the  atoms  are 
combined  into  molecules,  but  there  are  a  certain  number  of 
free  hydrogen  atoms  with  charge  e  and  half  as  many  of  the 

G.  E.  20 


306  ELECTRICITY  [CH.  XIX 

S04  groups  with  charge  —  2e.  These  free  atoms  and  groups 
are  known  as  ions.  When  the  electric  force  acts  the  free 
hydrogen  ions  are  carried  in  the  direction  of  the  force ;  some 
of  them  combine,  on  the  way,  with  the  free  sulphion  groups 
which  with  their  negative  charges  are  moving  in  the  opposite 
direction;  some  reach  the  negative  electrode  and  give  up  to 
it  their  positive  charge.  The  sulphion  groups  combine  with 
the  hydrogen  of  the  water  molecules,  thus  setting  free  oxygen 
ions  with  negative  charges  which  travel  to  the  positive 
plate;  in  this  way  we  can  explain  the  main  phenomena  of 
electrolysis. 

191.  Chemical  and  Electrical  Transformations 
in  a  Voltaic  Cell.  Let  us  now  consider  how  we  can  apply 
these  ideas  to  the  phenomena  of  the  cell. 

In  accordance  with  them  when  a  plate  of  pure  zinc  is 
placed  in  dilute  acid  the  zinc  begins  to  combine  with  the  acid 
and  form  zinc  sulphate.  Each  atom  of  zinc  carries  with  it 
its  charge  +  2e  and  leaves  the  zinc  plate  negatively  charged. 

We  here  assume  that  zinc  consists  of  an  equal  number  of  positively 
and  negatively  charged  atoms;  when  the  positive  atom  combines  with 
the  S04  the  negative  charge  is  set  free  on  the  zinc  plate. 

But  corresponding  to  each  negative  ion  SO4  which  has 
combined  with  the  zinc  two  positive  hydrogen  ions  have  been 
left  free  in  the  liquid,  these  are  attracted  to  the  negatively 
charged  zinc  and  form  a  coating  round  it,  and  a  double  sheet 
is  produced  about  the  zinc  consisting  of  the  negatively  charged 
zinc  ions  overlaid  on  the  outside  by  the  positive  hydrogen 
ions,  giving  rise  to  a  kind  of  molecular  condenser  in  which 
the  acid  is  the  positive  plate,  the"  zinc  the  negative. 

The  potential  of  the  zinc  thus  falls  below  that  of  the 
acid.  Hence  there  will  be  an  electric  force  tending  to  drive 
positively  charged  ions  from  the  acid  into  the  zinc,  counter- 
acting, that  is  to  say,  the  tendency  of  the  positive  zinc  ions  to 
dissolve  in  or  combine  with  the  acid.  Thus  the  electric  force 
due  to  this  double  layer  checks  the  solution  of  the  zinc,  and  it 
can  be  shewn  that  the  amount  of  zinc  which  can  be  thus 
dissolved  before  the  action  ceases  will  if  the  zinc  be  pure  be 
infinitesimal ;  we  could  not  expect  to  detect  it  by  any  known 


190-191] 


THE   VOLTAIC   CELL 


307 


method  of  analysis.     The  state  of  affairs  then  is  as  shewn  in 
Fig.  181. 


Acid 


Acid 


Fig.  181. 

The  amount  of  electricity  required  will  depend  on  the  distance 
between  the  two  layers  of  molecules;  taking  this  as  10~8  cm. — one 
hundred  millionth  of  a  centimetre — then  the  oxidation  of  5  '5  x  10~8 
grammes  of  zinc  per  square  centimetre  would  suffice  to  give  the  measured 
difference  of  potential. 

If  a  piece  of  copper  be  now  placed  in  the  acid  the  same 
kind  of  action  may  be  supposed  to  go  on,  but  if  we  suppose  the 
force  tending  to  make  the  copper  combine  with  the  acid  is  less 
strong  than  that  which  acts  in  the  case  of  the  zinc,  a  less  dense 
hydrogen  layer  will  be  required  to  produce  sufficient  electric 
force  to  balance  this  force,  and  the  potential  difference  between 
the  acid  and  the  copper  will  be  less  than  that  between  the 
acid  and  the  zinc.  Thus  the  zinc  will  be  at  a  lower  potential 
than  the  copper.  If  the  two  then  be  joined  by  a  wire,  positive 
electricity  passes  from  the  copper  to  the  zinc,  thus  reducing 
the  potential  difference  between  it  and  the  acid.  This  lessens 
the  electric  force  which  tends  to  prevent  the  positive  zinc  ions 
from  passing  into  solution  and  in  consequence  more  zinc  is 
dissolved,  at  the  same  time  the  transference  of  the  positive 
electricity  from  the  copper  to  the  zinc  reduces  the  potential 
of  the  copper  and  so  produces  a  further  accumulation  of  the 

20—2 


308 


ELECTRICITY 


[CH.  XIX 


positively  charged  hydrogen  ions  round  the  copper  and  thus 
the  current  is  kept  up. 

According  to  this  theory  of  the  cell,  then,  the  zinc  plate 
and  the  copper-wire  attached  to  it  are  when  the  circuit  is 
open  at  the  same  potential.  This  potential  is  lower  than 
that  of  the  acid  ;  the  copper  plate  also  is  lower  in  potential 
than  the  acid  but  above  the  zinc  plate,  the  difference  between 
it  and  the  zinc  plate  being  about  O8  volt,  and  this  constitutes 
the  E.M.F.  of  the  cell. 


Fig.  182. 

This  statement  of  the  action  of  the  cell  was  given  in 
Section  144,  and  the  distribution  of  potential  is  shewn  in 
Figs.  182  and  183. 


If  the  zinc  be  uot  pure  the  equilibrium  condition  is  not  reached ; 
local  differences  of  potential  are  produced  in  the  zinc  about  the  impurities, 
local  currents  are  set  up  and  the  zinc  is  continuously  dissolved. 


191-192] 


THE   VOLTAIC   CELL 


SOD 


192.     Volta's   Theory.     Contact   potential.     The 

explanation  of  the  action  of  the  cell  given  in  the  last  Section 
is  based  on  the  hypothesis,  that,  except  for  a  small  difference 
depending  on  temperature,  the  zinc  plate  of  a  cell  and  the 
copper-wire  attached  to  it  are,  when  the  circuit  is  open,  at 
the  same  potential. 

There  are  however  a  number  of  experiments  which  would 
appear  to  shew  that  two  metals  in  contact  are  at  different 
potentials.  Volta  was  the  first  to  observe  this.  In  some  of 
his  experiments  he  used  the  condensing  electroscope,  Fig.  184 
described  in  §  47.  For  this  purpose  the  plate  to  which  th} 
gold  leaves  are  attached  may  e 

be  made  of  copper  ;  the  upper 
surface  of  this  plate  is  covered 
with  a  thin  layer  of  shellac  or 
other  insulating  varnish,  the 
upper  plate  is  of  zinc  and 
is  carried  in  an  insulating 
handle.  The  upper  plate  is 
laid  on  the  lower,  being  sepa- 
rated from  it  by  the  varnish, 
and  contact  is  made  tempo- 
rarily between  the  backs  of 
the  two  plates  by  means  of  a 
wire. 

The  contact  is  then  broken 
and  the  zinc  plate  removed. 
When  this  is  done  the  gold  r 
leaves  diverge  and  on  testing 
their  charge  is  found  to  be 
negative. 

We  can  explain  this  by  supposing  that  a  small  potential 
difference  is  produced  by  the  contact  between  the  zinc  and 
the  copper,  the  copper  being  negative.  The  gold  leaves  are 
not  sensitive  enough  to  shew  this  directly,  but  the  two  plates 
constitute  a  condenser  of  large  capacity  and  thus  a  consider- 
able transference  of  positive  electricity  takes  place  from  the 
copper  to  the  zinc ;  on  removing  the  zinc  plate  the  negative 
electrification  left  on  the  copper  distributes  itself  over  the 
leaves  which  diverge  as  indicated.  If  however  the  experiment 


Fig.  184. 


310 


ELECTRICITY 


[CH.  XIX 


be  repeated,  but  contact  be  made  between  the  plates  by  means 
of  a  piece  of  thread  soaked  in  dilute  acid  which  is  afterwards 
removed,  no  such  potential  difference  shews  itself. 

These  results  led  Volta  to  construct  the  Voltaic  pile  shewn 
in  Fig.  185,  in  which  a  series  of  discs  of  copper,  zinc  and  moist 
flannel  are  strung  together  in  the  order 
copper,  flannel,  zinc,  copper,  flannel,  zinc, 
etc.,  and  in  which  a  potential  difference 
equal  to  0'8  multiplied  by  the  number 
of  couples  used  is  produced.  According 
to  the  theory  now  under  consideration 
any  three  adjacent  copper,  flannel  and 
zinc  discs  are  at  one  potential,  the  next 
copper  disc  is  at  lower  potential  than  the 
zinc  in  contact  with  it,  but  at  the  same 
potential  as  the  flannel  next  to  it  and 
the  zinc  beyond  this  flannel.  Thus  there 
is  a  drop  of  0*8  volt  between  any  pair 
of  consecutive  copper  discs. 

These  same  results  are  shewn  perhaps 
more  strikingly  by  the  aid  of  the  quadrant 
electrometer  and  this  enables  the  potential  differences  involved 
to  be  measured. 

Thus  a  plate  of  zinc  and  a  plate  of  copper  are  laid  side  by 
side  on  an  insulating  stand  and  connected  by  wires  to  the 
opposite  quadrants  of  an  electrometer.  On  connecting  the 
plates  by  a  wire  the  electrometer  needle  is  deflected  in  the 
direction,  and  to  the  amount,  which  indicate  that  the  zinc  is 
at  a  higher  potential  than  the  copper  by  0*8  volt. 

The  same  result  may  be  shewn  in  an  experiment  due  to 
Lord  Kelvin  thus  : — 

A  plate  of  zinc  and  a  plate  of  copper  are  laid  side  by  side 
in  a  horizontal  position  on  an  insulating  stand  as  in  Fig.  186, 
the  two  being  insulated  from  each  other.  An  electrometer 
needle  is  mounted  above  them  as  shewn  in  the  figure ;  the  axis 
of  the  needle  being  parallel  to  the  adjacent  edges  of  the  plates 
and  the  needle  is  electrified,  let  us  suppose  positively.  On 
connecting  the  plates  by  a  piece  of  wire  the  needle  is  deflected, 
being  attracted  to  the  copper  plate,  thus  this  plate  is  negative. 


Fig.  185. 


192] 


THE   VOLTAIC   CELL 


311 


If  the  .copper  wire  be  removed  and  a  drop  of  dilute  acid  placed 
between  the  plates  no  deflexion  takes  place. 


Fig.  186. 

Thus  it  would  appear  that  zinc  and  copper  in  contact  differ 
in  potential  by  about  0-8  volt  while  when  immersed  in  dilute 
acid  they  are  at  the  same  potential.  It  will  be  noticed  that 
this  potential  difference  O'S  volt  is  just  equal  to  the  E.M.F.  of 
a  zinc  copper  acid  cell.  According  then  to  this  theory  the 
action  of  the  cell  is  as  follows.  When  the  zinc  and  copper 
plates  are  immersed  in  the  acid  on  open  circuit  all  three  are 
at  the  same  potential.  A  copper  wire,  however,  connected  to 
the  zinc  plate  is  at  a  lower  potential  than  the  zinc,  at  a  lower 
potential  therefore  than  the  copper  plate.  When  then  the 
other  end  of  the  wire  is  connected  to  the  copper  plate  positive 
electricity  flows  from  the  plate  into  the  wire  lowering  the 


;  Copper: 


Fig.  187. 

potential  of  the  copper  plate  in  the  acid,  positive  electricity 
then  flows  from  the  zinc  plate  through  the  acid  to  the  copper 


312 


ELECTRICITY 


[CH.  XIX 


causing  decomposition  of  the  acid  and  liberating  the  energy 
required  to  maintain  the  current.  The  distribution  of  potential 
when  the  circuit  is  open  is  given  in  Fig.  187,  that  when  the 
current  is  flowing  is  shewn  in  Fig.  188,  the  main  difference 
of  potential  in  the  circuit  takes  place  at  the  copper-zinc 
junction. 


193.  Chemical  and  Contact  Theories.  In  order 
to  reconcile  these  views  we  may  notice  that  the  contact 
potential  experiments  are  ordinarily  made  in  air  or  with 
plates  which  have  been  exposed  to  air.  Now  the  active  agent  in 
the  action  between  the  zinc  and  acid  described  in  Section  190 
is  probably  the  oxygen  which  is  also  present  in  air ;  we  may 
assume  then  as  probable  that  the  same  action  tends  to  go  on 
between  the  zinc  and  the  air  as  between  the  zinc  and  the  acid 
— a  potential  difference  is  established  between  these  two. 

We  suppose — and  there  are  other  lines  of  argument  which 
lead  to  the  same  hypotheses — that  an  oxygen  molecule  in  air 
consists  of  two  atoms,  one  having  a  positive  charge  the  other  a 
negative ;  these  are  usually  combined  but  there  are  a  certain 
number  of  free  positive  and  negative  ions  present.  In  the 
neighbourhood  of  a  plate  of  zinc  negatively  charged  zinc  ions 
are  escaping  from  the  zinc,  these  combine  with  the  positive 
oxygen  ions  leaving  the  negative  oxygen  ions  to  form  with 
the  positively  charged  zinc  ions  a  double  sheet  over  the  zinc. 
In  the  same  way  a  potential  difference  is  established  between 
the  copper  and  the  air  near  it ;  thus  when  the  copper  and  zinc 
are  in  contact  they  are  at  the  same  potential,  but  the  air 


192-195] 


THE   VOLTAIC    CELL 


S13 


near  the  zinc  would  in  this  case  be  at  a  higher  potential  than 
that  near  the  copper.  Lines  of  force  pass  through  the  air 
from  the  positive  film  near  the  zinc  to  the  film  near  the  copper. 
Hence  a  positively  charged  needle  placed  near  is  repelled  from 
the  zinc  to  the  copper. 

194.  Contact     Experiments     in     a    Vacuum. 

Attempts  have  been  made  to  verify  this  by  conducting  the 
contact  experiment  in  a  vacuum  ;  the}7  have  however  failed 
and  calculation  shews  that,  in  any  vacuum  we  can  produce, 
there  must  be  a  number  of  oxygen  atoms  many  times  larger 
than  is  required  to  produce  the  double  layer  over  the  surface 
described  in  Section  190.  Hence  the  result  has  been  that  the 
potential  difference  observed  in  a  vacuum  between  copper  and 
zinc  is  the  same  as  that  found  in  air. 

By  replacing  the  oxygen  by  chlorine  a  change  in  the 
contact  difference  has  been  observed ;  it  is  extremely  difficult 
however  to  make  sure  that  this  was  not  due  to  actual  chemical 
action  occurring  between  the  chlorine  and  the  metals. 

195.  Acid-Metal    Contact.     When  the  two  metals 
are  connected  by  acid  they  are  really  at  different  potentials, 
but   the   electrometer   needle   is    not    deflected,    for   the   air 
near  each  of  the  two  metals  is  at  the  same  potential.     The 
potential  of  the  air  near  the  zinc  is  as  much  above  that  of 
the  zinc  as  is  the  potential  of  the  acid,  and  in  the  same  way 


Fig.  189. 

the  potentials  of  the  acid  and  of  the  air  near  the  copper  exceed 
that  of  the  copper  by  the  same  amount.  The  diagram  of 
potential  is  as  shewn  in  Fig.  189;  the  air  is  throughout  at 


314  ELECTRICITY  [CH.  XIX 

the  same  potential,  that  of  the  acid ;  so  far  as  the  metals  are 
concerned  there  is  110  field  of  force  and  the  electrometer  needle 
is  not  disturbed. 

196.  Contact  Potential.  The  contact  potential 
difference  described  as  existing  between  zinc  and  copper  can 
be  observed  in  a  similar  manner  between  any  pair  of  metals 
and  we  can  arrange  the  metals  as  was  done  by  Volta  in  a 
series  which  has  the  property  that  the  potential  in  air  of  any 
metal  in  the  list — or  as  we  probably  ought  to  express  it  the 
potential  of  the  air  near  any  metal  in  the  list — is  greater  than 
that  of  any  which  follows  it. 

The  following  is  such  a  list — zinc  lead  tin  iron  copper 
silver  gold  platinum  carbon — and  it  should  be  noted  that  the 
order  in  this  list  is  very  nearly  the  same  as  the  order  of  the 
heats  of  combination  between  the  metals  and  oxygen. 

The  potential  differences  between  the  consecutive  pairs  are 
given  in  the  following  Table  due  to  Ayrton  and  Perry. 

Zinc          ) 

•210 


Lead 

-099 

Tin 

(      -313 

Iron 

-146 

Copper 

j       -238 

Platinum  ' 

-113 

Carbon      ) 

The  difference  between  zinc  and  carbon  is  the  sum  of  these 
numbers  or  1. 

Moreover  it  is  shewn  by  experiments  that  if  we  have  three 
metals  A,  B,  C  arranged  so  that  A  is  in  contact  with  B  and  B 
with  C  the  difference  of  potential  between  A  and  C  is  exactly 
what  we  observe  when  A  and  C  are  put  into  connexion  directly, 
so  that  if  we  connect  to  C  a  second  piece  A  of  the  metal  A, 
then  A'  is  at  the  same  potential  as  A ;  hence,  if  A  and  A'  be 
joined  we  do  not  get  a  current  in  the  circuit. 


195-196] 


VOLTAIC    CELL 


315 


Let  us  for  example  suppose  that  there  is  a  rise  of  -5  volt 
between  A  and  B  and  a  further  rise  of  -8  between  B  and  C  so 
that  C  is  1  '3  volts  above  A ,  it  will  be  found  that  there  is  a  fall 
of  1-3  volts  between  C  and  A' ;  thus  A  and  A'  are  at  the  same 
potential,  if  they  be  joined  the  total  electromotive  force  round 
the  circuit  at  the  three  junctions  is  -5  +  -8  —  1-3  or  zero. 

On  the  chemical  theory  of  course  the  metals  when  in 
contact  are  all  at  the  same  potentials ;  the  differences  occur 
between  the  portions  of  air  in  the  immediate  neighbourhood  of 
the  respective  conductors.  The  distribution  of  potential  would 
be  as  in  Fig.  190  where  the  lower  line  gives  the  potentials  of 
the  metals,  the  upper  that  of  the  air  in  contact  with  them. 


Zinc 


1-3 


1 — Copper 


Fig.  190. 

It  has  been  tacitly  assumed  throughout  the  above  that  the 
conductors  are  throughout  at  the  same  temperature.  If  in 
a  circuit  of  two  metals  one  junction  is  maintained  at  a  higher 
temperature  than  the  other,  then  as  we  have  already  seen  a 
current  flows  round  the  circuit.  See  Section  182. 


316  ELECTRICITY  [CH.  XIX 


EXAMPLES  ON  VOLTAIC  ELECTRICITY. 

1.  If  the  electrochemical  equivalent  of  hydrogen  is  1'03  +  10~4  what 
are  the  electrochemical  equivalents  of  (1)  oxygen,  (2)  iron,  first  in  ferrous, 
second  in  ferric  salts  ? 

(Atomic  wt.  of  iron  =  56,  ferrous  iron  is  divalent,  ferric  iron  is  trivalent.) 

2.  If  a  current  of  4-8  amperes  will  decompose  8  grams  of  water 
in  5  hours,  what  is  the  electrochemical  equivalent  of  hydrogen  ? 

3.  If  the  electrochemical  equivalent  of  hydrogen  is  '0001  gr.,  how 
much  copper  will  be  deposited  in  one  hour  by  a  current  of  one  ampere  ? 

(Atomic  wt.  of  copper  =  63-4.) 

4.  If  1  ampere  deposits  4  gr.   of  silver  in  one   hour,  how  much 
hydrogen  will  be  liberated  by  half  an  ampere  in  a  week? 

(Equivalent  wt.  of  silver  =  108.) 

5.  A  current  passes  through  3  voltameters  in  series ;  one  contains 
a  solution  of  silver  nitrate,  the  second  a  solution  of  copper  sulphate,  and 
the  third  acidulated  water  ;  it  is  found  that  2-7  gr.  of  silver  are  deposited. 
Calculate  the  mass  of  copper,  hydrogen  and  oxygen  liberated,  having 
given  the  following  atomic  weights. 

(Hydrogen  =  1,  oxygen  =  16,  copper  =  63*4,  silver  =  108.) 

'•  6.  The  E.M.F.  of  a  given  battery  of  cells  of  total  internal  resistance 
5  ohms  is  10  volts.  The  poles  of  the  battery  are  joined  by  a  wire  of 
resistance  25  ohms.  Find  in  C.G.S.  measure  the  current  produced,  and 
the  difference  of  potential  between  the  poles. 

V  7.  The  E. M.F.  of  a  battery  on  open  circuit  is  4 -8  volts;  when 
producing  a  current  of  1'5  amperes  through  a  wire  the  difference  of 
potential  sinks  to  3  volts ;  find  the  resistance  of  the  wire  and  of  the 
battery. 

\J  8.  The  E.  M.F.  of  a  battery  on  open  circuit  is  10  volts.  When 
producing  a  current  of  5  amperes  the  difference  of  potential  between  its 
poles  is  8  volts,  find  its  resistance. 

/  9.  The  E.M.P.  of  a  battery  is  15  volts  and  its  total  internal  resistance 
is  6  ohms.  The  poles  of  the  battery  are  joined  by  a  wire  of  resistance 
30  ohms.  Find  the  current  produced  and  the  difference  of  potential 
between  the  poles. 

10.  A  cell  gives  a  current  of  1  ampere  when  its  terminals  are 
joined  by  a  wire  of  no  appreciable  resistance,  and  ^  ampere  when  joined 
by  a  wire  of  2  ohms  resistance ;  find  its  E.  M.  F.  and  its  resistance. 

Vll.    A  cell  sends  a  current  of  1  ampere  through  an  external  re- 
sistance of  1   ohm,  and  2  amperes  through  an  external  resistance  of 
ohm.     Find  its  E.  M.  F.  and  its  resistance. 


EXAMPLES    ON   VOLTAIC   ELECTKICITY  317 

12.  Account  for  the  fact  that  a  Leclanche  cell  will  send  a  greater 
current  through  a  long  thin  wire  than  a  Daniel  1's  cell,  while  if  the  wire 
is  short  and  thick  the  reverse  is  the  case. 

J  13.  A  cell  of  E.M.  F.  1  volt  sends  a  current  of  \  an  ampere  through  a 
galvanometer  whose  resistance  is  1  ohm.  What  current  would  be  sent 
by  3  such  cells  in  series  through  the  same  galvanometer? 

14.  You  have  12  Grove  cells,  each  giving  an  E.M.F.  of  1-9  volts 
and   having  a  resistance  of  £  ohm.     How  would  you  couple  them  so 
as  to  get  the  greatest  possible  current  through  a  resistance  of  £  ohm  ? 

15.  Two  equal  cells  when  connected  in  series  through  a  given  wire 
produce   a   current  of   10  amperes,  while  when   connected   in   multiple 
arc  through  the  same  wire  the  current  is  7*5  amperes.     Shew  that  the 
resistance  of  the  wire  is  2^  times  that  of  either  cell. 

16.  How  must  a  battery  of   10  cells,  each  having  a  resistance  of 
2  ohms,  be  arranged  so  as  to  give  the  largest  current  through  an  external 
resistance  of  6  ohms  ? 

17.  Three  cells  each  having  an  E.  M.  F.  of  1  volt  and  an  internal 
resistance  of  2  ohms  are  available ;    how  would  you  arrange  them  so 
as  to  obtain  the  greatest  current  through  a  circuit  of  2  ohms  resistance  ? 

18.  Ten  Daniell's  cells,  each  having  an  E.M.F.  of  1  volt  and  an 
internal   resistance  of  2  ohms,  are  arranged  in  2  parallel  groups,  the 
5  cells  of  each  group  being  placed  in  series.     Calculate  the  current  which 
they  will  send  through  a  galvanometer  having  a  resistance  of  30  ohms, 
with  its  terminals  connected  through  a  coil  whose  resistance  is  6  ohms. 

19.  Ten   voltaic   cells,   each   of   E.M.F.    1-75   volts   and  resistance 
•75  ohm,  are  joined  in  series  and  the  circuit  completed   by  a  wire  of 
resistance  12-5  ohms.     Find   (1)  the  strength  of  the  current,  (2)  the 
quantity  of  electricity  that  passes  any  section  of  the  circuit  per  minute, 
(3)  the  difference  of  potential  at  the  terminals  of  the  battery. 

20.  You  are  given  48  cells  each  of  E.M.F.  1-8  volts  and  resistance 
•3  ohm.     How  would  you  arrange  them  to  produce  the  greatest  current 
in  a  circuit  of  5  ohms  resistance  ? 

V  21.  A  Daniell's  cell  has  E.M.F.  !•!  volts  and  internal  resistance 
•5  ohm.  Its  terminals  are  joined  by  2  wires  arranged  in  parallel  of 
resistances  1  and  1-5  ohms  respectively.  Find  the  current  in  each  wire 
and  in  the  cell. 

y  22.  You  are  given  a  battery  of  E.  M.  F.  5  volts  and  resistance  2  ohms, 
a  coil  of  wire  of  resistance  8  ohms  and  a  galvanometer  of  resistance 
5  ohms.  Calculate  the  current  (a)  through  the  cell  and  (b)  through  the 
galvanometer  when  (1)  all  three  are  in  series,  (2)  the  poles  of  the  battery 
are  connected  to  the  galvanometer  and  the  coil  in  multiple  arc. 

23.  Two  electrolytic  cells  each  containing  copper  sulphate  the  re- 
sistance of  which  is  very  high  compared  with  all  other  resistances  in  the 
circuit  are  placed  first  in  series,  and  secondly  in  multiple  arc ;  compare 
the  total  quantities  of  salt  decomposed  in  the  two  cases. 


318  ELECTRICITY  [CH.  XIX 

24.  The  poles  of  a  Daniell's  cell  (E.M.F.  1-08  volts,  resistance  -5  ohm) 
are  connected  in  multiple  arc  by  two  wires  ACS  and  ADB  ;  the  resistances 
of  ADB  and  the  part  AC  are  each  1  ohm,  what  must  be  the  resistance 
of  the  part  CB  in  order  that  the  difference  of  potential  between  A  and  G 
may  be  -01  volt  ? 

^  25.  Two  points  A  and  B  are  joined  by  two  conductors  ADB  and 
ACB.  The  one  ADB  has  a  resistance  of  1  ohm,  the  other  ACB  is 
98  ohms  between  A  and  C  and  1  ohm  between  C  and  B.  If  a  current 
of  1  ampere  enters  the  system  at  A  and  leaves  it  at  B,  find  the  difference 
in  potential  between  C  and  B. 

26.  A  current  passes  through  a  water  voltameter  A,  and  then  divides 
and  passes  partly  through  a  copper  voltameter  B  and  partly  through 
a  silver  voltameter  C,  B  and  C  being  arranged  in  parallel.  If  1  gr.  of 
copper  is  deposited  in  B  and  2  gr.  of  silver  in  C,  how  many  gr.  of 
hydrogen  will  be  set  free  in  A  ? 

(Atomic  wt.  of  copper =63-4,  of  silver  =  108. ) 

J  27.  Two  wires  of  resistances  50  and  10  ohms  respectively  connect 
two  points.  What  is  the  effective  resistance  of  these  two  wires  when 
in  multiple  arc  ? 

\  28.  If  two  conductors  of  3  ohms  and  4  ohms  resistance  respectively 
are  joined  in  parallel  what  is  their  combined  resistance  ? 

J  29.  Enumerate  all  the  resistances  that  can  be  obtained  from  3  coils 
of  resistances  2,  4  and  6  ohms  respectively  by  the  various  ways  in  which 
they  may  be  connected,  all  three  coils  being  always  in  use. 

30.  A  galvanometer  having  a  resistance  of  5000  ohms  is  shunted 
with  100  ohms.    A  certain  deflexion  of  the  galvanometer  is  obtained  with  a 
battery  of  constant  E.M.  F.  when  the  resistance  of  the  rest  of  the  circuit 
is  2000  ohms.     What  additional  resistance  must  be  inserted  to  produce 
the  same  deflexion  when  the  shunt  is  removed? 

31.  A  battery  is  connected  up  by  thick  wires  to  a  galvanometer  and 
the  current  is  observed.     On  shunting  the  galvanometer  with  ^T  of  its 
own  resistance  the  current  is  halved  ;    shew  that  the  resistance  of  the 
galvanometer  is  20  times  that  of  the  battery. 

32.  If  a  battery  of  very  low  internal  resistance  R  is  connected  with 
the  terminals  of  a  galvanometer,  the  deflexion  is  almost  unaltered  when 
the  instrument  is  shunted ;  but  if  the  internal  resistance  of  the  battery 
is  high,  the  alteration  is  very  considerable.     Explain  this. 

33.  A  battery  is  connected  by  short  thick  wires  to  a  galvanometer 
and  the  deflexion  noted.     The  galvanometer  is  then   shunted  with  J 
of  its  own  resistance,  and  on  again  connecting  with  the  battery  the 
current  through  the  galvanometer  is  observed  to  have  half  its  former 
value.      Shew  that  the  resistance  of  the  battery  is  half  that  of  the 
galvanometer. 

34.  When  2  coils  of  resistances  10  ohms  and  5  ohms  respectively 
are  connected  up  in  series  with  a  galvanometer  and  a  battery  of  negligible 


EXAMPLES   ON   VOLTAIC   ELECTRICITY  319 

resistance  the  current  indicated  is  -2  ampere  ;  when  the  coils  are  in 
parallel  the  current  is  -35  ampere.  Calculate  the  resistance  of  the 
galvanometer. 

35.  A  reflecting  galvanometer  has  a  resistance  of  100  ohms  and  is 
shunted  with  TV  ohm  ;  a  battery  of  very  low  internal  resistance  and  E.M.F. 
of  2  volts  is  put  in  series  with  it  and  10,000  ohms.     The  scale  deflexion 
observed  is  150  divisions  ;  find  the  sensibility. 

36.  A  galvanometer  of  100  ohms  resistance  is  placed  in  series  with 
a  resistance  box  of  45  ohms  and  with  a  battery  whose  E.M.F.  is  1-5  volts 
and  resistance  5  ohms,  and  the  deflexion  is  observed.     The  resistance 
box  is  then  short  circuited  and  the  same  deflexion  as  before  is  produced 
by  shunting  the  galvanometer.     Find  the  resistance  of  the  shunt. 

37.  The  resistance  of  a  piece  of  wire  1  mm.  radius  and  15'7  metres 
long  is  1  ohm,  find  its  specific  resistance. 


38.  The   specific  gravity   of  silver  is   10'5.     The  resistance  of  a 
silver  wire  100  cm.  long  and  1  gr.  in  weight  is  -1689  ohm.     Shew  that 
the  specific  resistance  of  silver  is  1609  using  absolute  units. 

39.  A   wire  is   stretched  uniformly  until    its    length    is  doubled. 
Compare  its  resistance  before  and  after  stretching. 

40.  A  uniform  copper  wire,  whose  resistance  is  12  ohms,  is  bent 
into  the  form  of  a  square  and  the  ends  soldered  ;  the  poles  of  a  battery, 
whose  resistance  is  3  ohms,  are  joined  at  two  opposite  corners  A  and  C 
of  the  square  ;   in  what  ratio  will  the  strength  of  the  current  flowing 
along  each  side  of  the  square  be  altered  by  joining  A  and  C  by  a  straight 
piece  of  the  same  copper  wire  ? 

41.  A  battery  when  joined  to  a  tangent  galvanometer  of  10  ohms 
resistance  gives  a  deflexion  of  60°.     If  a  resistance  of  20  ohms  is  inserted 
in  the  circuit  the  deflexion  falls  to  45°.     What  is  the  resistance  of  the 
battery  ? 

42.  Two   currents  passed  in  turn  round  a  tangent  galvanometer 
produce  deflexions  of  30°  and  60°  respectively.    Compare  the  strengths  of 
the  two  currents. 

43.  A  tangent  galvanometer  has  10  turns  of  wire  wound  in  a  groove 
of  radius  20  cm.  ;    what  current  (expressed  in  amperes)  will  give  it  a 
deflexion  of  30°? 

(H=-18  C.G.S.  unit.) 

44.  A  wire  is  coiled  into  a  circle  of  10  turns  and  used  as  the  coil  of 
a  tangent  galvanometer.   On  passing  a  current  of  1  ampere  the  deflexion 
is  45°.     Find  the  radius  of  the  circle. 

(H=-18  C.G.S.  unit.) 


320  ELECTRICITY  [CH.  XIX 

45.  A  current  of  5  amperes  flows  in  a  circular  wire  of  10  cm.  radius. 
How  many  turns  of   wire  are  there  in  the  coil  if  the  strength  of  the 
magnetic  field  at  the  centre  is  1  dyne  ? 

46.  The  coil  of  a  tangent  galvanometer  is  10  cm.  in  radius  ;   how 
many  turns  of  wire  must  be  wound  on  if  a  current  of  -01  ampere  is  to 
produce  a  deflexion  of  about  45°  ? 

(H=-18  c.o. s.  unit.) 

47.  Two  Daniell's  cells  give  equal  deflexions  on  a  quadrant  electro- 
meter, but  quite  different  deflexions  when  connected  with  a  low  resistance 
galvanometer.     What  do  you  suppose  is  the  cause  of  the  difference  ? 

48.  A  tangent  galvanometer  has  a  coil  of  10  turns  of  wire  with  a 
radius  of  10  crn.     What  mass  of  copper  will  be  deposited  from  a  cupric 
salt  in  half  an  hour  by  a  current  which  deflects  the  needle  through  45°  ? 

(H=-18  c.o.s.  unit.      ?r  =  3-14.     1  ampere   deposits   -000328  gr.   of 
copper  in  one  second.) 

49.  Two  copper  plates  are  immersed  in  a  solution  of  copper  sulphate 
and  a  current  passed  through  them  and  a  tangent  galvanometer.     The 
deflexion  of  the  galvanometer  is  45°,  and  after  an  hour  it  is  found  that 
216  milligrammes  of  copper  have  been  deposited  on  one  plate;  having 
given  that  a  current  of  1  ampere  deposits  19*8  milligrammes  per  minute, 
deduce  the  reduction  factor  of  the  galvanometer. 

50.  An  arrangement  of  resistance  keys  and  connecting  wires  is  made 
for  the  purpose  of  determining  an  electrical  resistance.     If  when  the 
galvanometer  circuit  is  made,  the  battery  key  being  open,  a  deflexion  of 
the  galvanometer  is  produced,  by  what  experiments  would  you  determine 
whether  the  deflexion  is  due  to  (1)  leakage  through  the  battery  key  or 
(2)  an  E.M.F.  in  the  circuit  independent  of  the  battery? 

51.  The  resistance  of  a   coil   of  copper  wire  is  determined  by  a 
Wheatstone's  Bridge  box  when  the  temperature  of  the  air  is  20°  C.  and  is 
found  to  have  the  value  20 '25  ohms.     Calculate  its  true  value  at  0°  C.,  the 
coils  of  the  box  being  of  german  silver  and  correct  at  15°  C. 

Coefficient  of  increase  of  resistance  per  1°  C.  for  copper  is  -0038 

„  „  ,,  ,,  german  silver  is  -0004. 

52.  A  careless  observer  in  setting  up  his  apparatus  for  the  measure- 
ment of  the  resistance  of  a  coil  by  means  of  the  Wheatstone  Bridge 
neglects  to  clean  the  ends  of  the  wires  by  which  the  connexions  are  made 
between  the  coil,  galvanometer  and  battery  and  the  rest  of  the  apparatus. 
Consider  the  effect  of  the  neglect  of  these  precautions  upon  (1)  the  result, 
(2)  the  sensitiveness  of  the  method. 

53.  A  wire  whose  resistance  was  to  be  determined  was  placed  in  a 
Wheatstone's  Bridge,  the  fixed  ratio  arms  of  which  were  10  and  100  ohms 
respectively ;    balance   was    obtained    when    the    adjustable   coils   were 
arranged  to  give  a  resistance  of  467  ohms.     What  was  the  value  of  the 
resistance  of  the  coil  under  examination? 


EXAMPLES   ON   VOLTAIC   ELECTRICITY  321 

54.  If  a  stretched  and  graduated  wire  whose  resistance  is  5  ohms  is 
connected  to  a  battery  whose  E.M.F.  is  3-1  volts  and  internal  resistance 
1-2  ohms,  what  is  the  limit  of  the  E.M.F.'S  which  can  be  compared  by 
means  of  it,  using  the  potentiometer  method? 

55.  A  fine  wire  is  placed  in   100  gr.  of  water   in   a   light   copper 
calorimeter  and  2  amperes  are  passed  through  it,  an  E.  M.  F.  of  14  volts 
being  maintained  between  the  ends  of  the  wire.     Calculate  the  rise  of 
temperature  in  10  minutes. 

(J=  4-2  xlO7  ergs.) 

56.  Through  a  coil  of  wire  which  is  immersed  in  water  in  a  small 
calorimeter  a  current  of  2  amperes  is  passed ;  the  calorimeter  contains 
200  gr.  of  water  at  a  temperature  of  15°  C. ;  at  the  end  of  5  minutes  it  is 
found  that  the  temperature  has  risen  to  18°  C.,  find  the  value  of  the 
electrical  resistance  of  the  coil. 

57.  A  platinum  wire  has  a  resistance  of  half  an  ohm.     You  are 
provided  with  8  Daniell's  cells  each  of  1  ohm  resistance.     How  would  you 
arrange  them  in  order  to  make  the  wire  as  hot  as  possible? 

58.  A  battery  of  E.M.F.  6  volts  and  resistance  2  ohms  is  connected  to 
two  wires  in  parallel  of  1  and  3  ohms  respectively.     Find  the  current  and 
the  heat  developed  per  second  in  each  branch. 

59.  The  E.  M.  F.  of  a  constant  battery  is  3  volts  ;  its  internal  resistance 
is   1  ohm.     The  battery  sends  its  current  in  the  first  place  through  a 
copper  wire  of  3  ohms'  resistance.     The  wire  is  then  removed  and  an 
electrolytic  cell  containing  dilute  sulphuric  acid,  also  of  3  ohms'  resistance, 
substituted  in  its  place..    If  it  require  an  E.M.F.  of  1*1  volts  to  electrolyse 
dilute  sulphuric  acid,  compare  the  heat  generated  per  minute  in  the  acid 
with  that  generated  in  the  wire. 

60.  A  current  is  passed  through  a  thin  wire  enclosed  in  a  calorimeter, 
and   through   a   copper   voltameter   arranged   in    series  with    the  wire. 
Calculate  the  resistance  of  the  wire  from  the  following  data : — 

Time  for  which  current  is  passed =20  minutes. 

Weight  of  copper  deposited    =  €3  gram. 

„       ,,    water  in  the  calorimeter =  100grams. 

Water  equivalent  of  the  calorimeter =  10  grams. 

Rise  of  temperature  corrected  for  radiation  losses...  =20°  C. 

Mechanical  equivalent  of  heat    =4*2  x  107ergs. 

Electrochemical  equivalent  of  copper  =  '0033  gram  per  c. a. s. 

unit  of  current. 

61.  What  will  be  the  ratio  of  the  currents  which  will  keep  two  wires 
of  the  'same  material  heated  to  the  same  temperature  if  the  radius  of  one 
wire  is  double  that  of  the  other  ? 

G.  E.  21 


322  ELECTRICITY  [CH.  XIX 

62.  If   the  loss  of  heat  from  a  wire  per  unit  area  of  surface  by 
radiation  and  convection  be  proportional  to  the  excess  of  its  temperature 
above  that  of  the  surrounding  air,  shew  that,  if  the  same  current  is  sent 
through  wires  of  the  same  material,  the  elevation  of  temperature  is  inversely 
proportional  to  the  cube  of  the  radius  of  the  wire. 

63.  A.  copper  wire    '02  cm.   in    diameter,   carrying    a   current   of 
1  ampere,  is  found  to  reach  a  steady  maximum  temperature  of  100°  C. 
Taking  the  specific  resistance  of  copper  at  100°  C.  as  2-1  x  10~6  ohms 
per  centimetre   cube,  and   the  value  of  J  as  4'2  x  107,   calculate  how 
many  units   of  heat   are   emitted  per   second    by  1  sq.  cm.   of  copper 
surface  at  100°  C. 

64.  A  battery  of  E.M.  F.  E  and  internal  resistance  B  drives  a  current 
through  a  wire  of  resistance  R.    Shew  that  the  heat  produced  in  the  wire 
in  a  given  time  is  a  maximum  when  R  =  B. 

65.  Find  the  horse-power  required  to  light  75  incandescent  lamps 
each  taking  \  ampere  and  requiring  100  volts  at  its  terminals.     If  one 
of  these  lamps  were  immersed  in  2  litres  of  water  and  then  lighted,  how 
fast  would  the  temperature  of  the  water  rise  ? 

(A  horse-power  =  746  watts  =  746  x  107  ergs  per  sec.) 

66.  Power  to  the  extent  of  100,000  watts  has  to  be  carried  to  a 
distance  of  5000  metres  with  a  loss  not  exceeding  5%.     Compare  the 
cost  of  the  copper  mains  if  the  current  has  a  voltage  of  100  with  their 
cost  if  the  voltage  is  raised  to  2000. 

67.  A  difference  of  potential  of  100  volts  is  maintained  between  the 
terminals  of  a  dynamo  machine  supplying  an  installation  of  100  lamps ; 
the  current  in  each  lamp  is  '65  of  an   ampere  and  the  resistance  of 
the  leads  to  each   lamp   is   1  ohm.     Calculate  the   amount  of  energy 
supplied  by  the  machine  per  hour  and  the  amount  wasted  in  the  leads. 

68.  A  pair  of  copper  bars  are  required  to  transmit  200  amperes  from 
a  dynamo  to  a  motor  at  1000  metres  distance.    The  E.M.F.  at  the  dynamo 
is  100  volts  and  the  E.M.F.  at  the  motor  is  not  to  be  less  than  96  volts. 
If  the  specific  resistance  of  copper  is  1-6  x  10~6  ohms  per  centimetre  cube, 
calculate  the  least  section  each  bar  may  have. 

69.  A.  current  is  passed  across  the  junction  of  two  metals,  and  it  is 
found  that  heat  is  evolved  at  this  junction.     How  would  you  distinguish 
between  the  heat  generated  by  overcoming  the  electrical  resistance  at  the 
junction  and  the  heat  due  to  the  Peltier  effect? 

70.  A  strong  current  is  sent  for  a  short  time  across  the  junction  of 
a  bar  of  antimony  and  a  bar  of  bismuth.     A  galvanometer  is  then  placed 
in  the  circuit  instead  of  the  battery  and  it  is  found  that  the  galvanometer 
needle  is  deflected.     Explain  this  phenomenon. 


CHAPTER  XX. 

ELECTROMAGNETISM. 

197.     Magnetic    Force    due   to   a    Current.     We 

have  already  seen  (§  112)  that  the  space  in  the  neighbourhood 
of  a  current  is  traversed  by  lines  of  magnetic  force — it  is  a  field 
of  force  —and  we  have  discussed  in  the  case  of  a  galvanometer 
how  this  fact  may  be  used  to  measure  a  current.  Reference 
has  also  been  made  to  the  magnetisation  of  iron  by  a  current. 
These  facts  are  of  the  greatest  importance  in  the  modern 
applications  of  electricity  and  we  must  now  consider  them 
more  closely. 

We  can  trace  experimentally  in  various  ways  the  lines 
of  force  due  to  a  current  in  a  coil  of  wire  ;  thus  take  a  circular 
coil  and  fix  it  so  that  its  plane  is  vertical ;  adjust  a  horizontal 
sheet  of  cardboard  as  shewn  in  Fig.  191  so  that  the  axis  of 
the  coil  may  lie  in  the  sheet  of  card  and  sprinkle  iron  filings 
over  the  card.  On  passing  a  current  through  the  coil  and 
tapping  the  card  the  iron  filings  arrange  themselves  along  the 
lines  of  magnetic  force  in  the  plane  of  the  coil. 

Or  again  if  a  small  compass  needle  be  placed  on  the  card 
the  needle  will  set  itself  when  the  current  passes  so  as  to  be 
tangential  to  the  line  of  force  through  its  centre  ;  the  direction 
of  the  line  of  force  will  pass  through  the  needle  from  its  south 
pole  to  its  north. 

The  lines  of  force  start  from  one  face  of  the  coil  and 
travelling  outwards  gradually  curve  round  and.  enter  the 
other  face  of  the  coil.  In  this  they  resemble  the  lines  of 
force  from  a  magnet  which  start  outwards  from  the  north 
pole  and  passing  round  enter  again  at  the  south  pole. 

21—2 


324 


ELECTRICITY 


[CH.  XX 


It  must  be  remembered  in  these  and  similar  observations 
that  the  lines  of  force  observed  are  due  to  the  resultant  action 
of  the  current  in  the  coil  and  the  earth's  magnetism.  In. 
Fig.  191  the  lines  are  drawn  as  undisturbed  bj  the  earth's 
force. 


Fig.  191. 

We  may  shew  as  the  result  both  of  theory  and  experiment 
that  if  the  coil  be  small  the  distribution  of  magnetic  force 
at  some  distance  away  is  the  same  as  that  due  to  a  small 
magnet  placed  with  its  centre  at  the  centre  of  the  coil  and  its 
axis  at  right  angles  to  the  plane  of  the  coil.  Moreover  if  the 
total  area  of  the  windings  of  the  coil  be  A  square  centimetres 
and  the  strength  of  the  current  in  the  wire  be  i  C.G.S.  units, 
then  the  magnetic  moment  of  the  equivalent  magnet  can  be 
shewn  to  be  Ai. 

In  estimating  A  the  number  of  turns  of  wire  on  the  coil  has  to  be 
taken  into  account;  it  is  the  sum  of  the  areas  of  the  individual  turns; 
thus  if  the  coil  consist  of  n  turns  each  of  area  a  the  value  of  A  is  na. 


197-198] 


E  LECTROM  AGNETISM 


325 


198.  Magnetic  Shells.  If  however  the  coil  cannot 
be  treated  as  very  small  we  cannot  represent  its  action  by 
that  of  a  single  small  magnet,  we  have  to  suppose  the  area  of 
the  coil  divided  into  a  number  of  small  equal  elements  or  parts 
and  place  at  the  centre  of  each  element  a  small  magnet  with 
its  axis  at  right  angles  to  the  plane  of  the  coil.  These  small 
magnets  are  all  equal  and  their  north  poles  all  point  in  the 
same  direction.  Moreover  if  there  be  n  of  them  and  if  A  be 
the  total  area  of  the  coil  and  i  the  current,  then  the  magnetic 
moment  of  each  is  Aijn. 

Such  a  distribution  of  magnets  constitute  if  they  are 
placed  quite  close  together  what  is  known  as  a  magnetic  shell. 
We  may  look  upon  them  as  forming  a  sheet  of  iron  or 
magnetic  material  bounded  by  the  coil ;  one  side  of  this  sheet 
will  be  coated  with  north  poles,  the  other  side  with  south 
poles.  The  distribution  of  magnetic  force  due  to  the  coil  will 
then  be  the  same  as  that  due  to  the  shell. 

It  is  important  to  determine  readily  which  face  of  the  coil 
is  to  be  treated  as  the  north-pointing  or  positive  face.  The 
right-handed  screw  rule  given  in  Section  112  enables  us  to  do 
this.  Imagine  the  right  hand  placed  on  the  wire  as  in  Fig.  192 
so  that  the  current  is  passing  from 
the  wrist  out  at  the  fingers  and  let 
the  extended  thumb  be  within  the 
area  formed  by  the  coil ;  then  the 
back  of  the  hand  represents  the 
north  pole  ends  of  the  equivalent 
magnets — the  north  face  of  the  coil 
we  may  call  it.  This  is  obvious  if 
we  remember  that  a  north  pole 
placed  at  the  end  of  the  thumb 
would  move  round  the  current  in 
the  same  direction  as  the  hand  must 
be  turned  to  screw  in  a  right- 
handed  screw  whose  point  coincides 
with  the  fingers  and  is  to  be  moved 
in  the  direction  in  which  the  current  is  travelling.  To  secure 
this  the  direction  of  motion  of  the  thumb  must  start  out- 
wards from  the  back  of  the  hand  ;  thus  the  lines  of  force 
issue  from  the  back  of  the  hand  placed  in  the  position 


Fig.  192. 


326 


ELECTRICITY 


[CH.  XX 


described  and  enter  at  the  front.      The  back  of  the  hand  is 
the  north  face  of  the  coil. 

Another  rule  may  be  given  which  is  sometimes  more 
convenient 

Place  a  watch  with  its  face  towards  the  observer  and 
coincident  with  the  plane  of  the  coil. 

If  the  current  circulates  in  the  coil  in  the  same  direction  as 
the  hands  of  the  watch  move  then  the  south  face  of  the  coil  is 
towards  the  observer,  Fig.  193  ;  if  the  direction  of  the  current 
is  opposite  to  that  of  the  motion  of  the  hands  the  north  face  is 
towards  the  observer,  Fig.  194. 


199.  Action  of  a  Magnet  on  a  Current.  If  a  coil 
of  wire  carrying  a  current  behaves  like  a  magnetic  shell  and 
produces  a  magnetic  field  in  its  neighbourhood,  then  if 
another  coil  or  a  magnet  be  brought  near  there  will  be 
magnetic  force  between  the  two.  It  was  by  studying 
experimentally  the  forces  between  a  magnet  and  a  coil  carrying 
a  current  that  the  equivalence  of  a  current  and  a  magnetic 
shell  was  established. 

Thus  let  a  coil  consisting  of  a  single  turn  of  wire  be 
supported  so  that  it  can  rotate  readily  about  a  vertical  axis. 
This  is  done  in  Ampere's  stand  by  balancing  the  coil  on  a  steel 
point  which  rests  in  a  cup  at  the  top  of  a  vertical  axis.  One 


198-199] 


ELECTROMAGN  ETISM 


327 


end  of  the  coil  is  connected  to  this  point.  A  second  annular 
cup  surrounds  the  first  but  is  insulated  from  it  and  both  cups 
contain  mercury.  The  second  end  of  the  coil  is  connected  to 
a  needle  which  dips  into  the  mercury  in  the  annular  cup  and 
maintains  electric  contact  between  it  and  the  cup  as  the  coil 
rotates  round  the  central  pivot;  if  then  one  pole  of  the  battery 
be  connected  to  the  outer  cup  and  the  other  pole  to  the  central 
cup  the  current  passes  through  the  coil  which  is  free  to  move 
round  a  vertical  axis. 

The  arrangement  is   shewn  in  Fig.   195,   where  A  is  the 
inner  cup,   G  the  outer. 


Fig.  195. 

AtiCDEFG  is  the  wire  balancing  on  a  pivot  at  A.  There 
is  a  bend  in  the  wire  at  D  to  allow  it  to  clear  the  support. 
If  a  current  be  passed  through  the  coil  from  A  to  G  it  will  be 
found  that  the  face  A  BCEFG  becomes  a  north  face  ;  the  coil 
sets  itself  with  its  plane  pointing  east  and  west ;  the  upper 
surface  of  the  paper  will  become  a  north  pole.  This  can 
be  also  tested  by  bringing  the  north  pole  of  a  magnet  near 
the  coil,  the  face  in  question  will  be  repelled,  the  opposite  face 
attracted. 


328 


ELECTRICITY 


[CH.  XX 


This  action  of  course  is  the  converse  of  that  made  use  of  in 
the  ordinary  galvanometer  already  described  in  which  the  coil 
is  fixed  and  the  magnet  moves. 

200.  Electromagnetic  Action  between  two 
Currents.  Again,  if  a  second  coil  carrying  a  current  is 
brought  near  the  suspended  coil,  forces  of  attraction  or 
repulsion  come  into  play.  The  direction  of  these  forces  may 
be  best  determined  by  considering  the  action  between  the  two 
magnets  to  which  they  are  equivalent ;  the  moveable  coil  will 
tend  to  set  itself  so  that  the  axes  of  the  magnets  to  which  it  is 
equivalent  lie  as  nearly  as  may  be  along  the  lines  of  force  due 
to  the  second  coil;  the  direction  of  the  axis  of  a  magnet,  it 
must  be  remembered,  runs  from  its  south  pole  to  its  north 
pole.  Thus  if  the  two  coils  be  side  by  side  as  in  Fig.  196  a 


B 


>   T**J  I 

v           ^ 

f^''      A             / 

- 

M            C                            C 
Fig.  196  a.  ^ 

the  coil  KM  being  fixed  and  BD  moveable  and  the  current 
circulate  in  the  same  direction  round  the  two  so  that  it  runs 
downwards  in  BC,  and  upwards  in  MN,  a  side  of  the  fixed 
coil  adjacent  and  parallel  to  £C,  then  the  north  faces  of  both 


N 


B 


t                    / 

\           / 

A                 N 

M           D                          C 

Fig.  196  b. 

coils  are  on  the  upper  side  of  the  paper,  the  lines  of  force 
from  the  fixed  coil  KM  pass  through  the  moveable  coil  BD 


199-200] 


ELECTROMAGNETISM 


329 


downwards,  that  is  from  north  to  south;  thus  the.moveable 
coil  will  tend  to  set  itself  in  the  reverse  position  so  that  the 
lines  of  force  may  traverse  it  from  south  to  north,  hence  ED 
becomes  adjacent  to  NM  and  the  currents  in  the  two  parallel 
wires  run  in  the  same  direction  as  in  Fig.  196  b. 

Thus  two  parallel  wires  carrying  currents  repel  each  other 
when  the  currents  are  in  opposite  directions,  they  attract  when 
the  currents  run  in  the  same  direction. 

Again  let  the  fixed  coil  be  placed  below  the  moveable  one, 
the  planes  of  both  being  vertical  but  not  coincident  and  the 
sides  of  the  coils  horizontal  and  vertical. 

Then  looking  down  on  the  coils  their  projections  in  a 
horizontal  plane  would  be  EOB  and  LOM,  Fig.  197*7,  while 
the  current  runs  from  L  to  M  and  from  E  to  B,  where  LM 
is  the  upper  side  of  the  fixed 
circuit,  EB  the  lower  side  of  the 
moveable  one. 

The  north  faces  of  both 
circuits  are  to  the  right.  The 
lines  therefore  from  the  upper 
fixed  circuit  pass  through  the 
lower  moveable  circuit  from 
north  to  south,  thus  the  lower 
circuit  tends  to  reverse  its  posi- 
tion and  to  set  itself  so  that 
the  lines  traverse  it  from  south 
to  north,  the  two  wires  tend  to 
become  parallel  while  the  currents  in  the  two  parallel  wires 
run  in  the  same  direction,  as  in  Fig.  197  b. 

Thus  when  the  currents  run  in  two  adjacent  wires  as  in 
Fig.  197  a  the  acute  angle  between  the  wires  is  increased  by 
the  action  between  the  currents,  if  the  current  were  reversed 
in  one  wire,  say  LAf,  the  acute  angle  would  be  decreased. 

In  the  first  case',  limiting  ourselves  to  the  portions  of  the 
wire  below  the  point  0,  the  current  in  one  wire  is  running 
towards  the  angle,  in  the  other  wire  it  is  running  from  the 
angle  ;  in  the  second  case  witli  BE  reversed  the  current  in 
both  wires  is  running  from  the  angle. 

In  all  these  cases  the  moveable  circuit  tends  to  set  itself 


Fig.l97a. 


Fig.  197  6. 


330 


ELECTRICITY 


[CH.  XX 


so  that  the  number  of  lines  of  force  which  traverse  it  from 
south  to  north  should  be  as  great  as  possible. 

2O1.     Solenoid.     By  making  the  coil  in  the  form  of  a 
spiral  having  a  considerable  number  of  turns  as  in  Fig.   198 


Fig.  198. 

instead  of  the  single  turn  shewn  in  Fig.  195  its  magnetic 
moment  is  greatly  increased.  Each  spire  of  the  coil  is 
equivalent  to  a  short  magnet  of  small  moment,  or  more  strictly 
to  a  magnetic  shell  with  its  axis  lying  along  the  axis  of  the 
coil,  and  this  series  of  magnetic  shells  constitutes,  a  magnet 
of  the  same  length  as  the  axis  of  the  spiral. 

If  the  spiral  be  balanced  at  its  centre  on  the  ampere  stand, 
when  the  current  passes,  it  sets  itself  with  its  axis  north  and 
south.  To  an  observer  looking  at  the  north  pole  of  the  coil 
the  current  appears  to  circulate  as  in  Fig.  199  a,  in  the 
opposite  direction  to  the  motion  of  the  hands  of  a  watch,  to 
an  observer  looking  at  the  south  pole  the  direction  of  the 


Fig.  199  a. 


Fig.  1996. 


current  appears   to  circulate    as   in   Fig.    199  6,  in  the  same 
direction  as  the  hands  of  the  watch  move. 

If  the  spiral  be  sufficiently  long  the  magnetic  force  within 
it  is  very  approximately  uniform,  the  lines  of  force  are  parallel 


200-203] 


ELECTROMAGNETISM 


331 


to  the  axis  of  the  spiral,  and  it  can  be  shewn  that  at  a 
sufficient  distance  from  the  ends  this  force  is  equal  to  4mri/l 
where  i  is  the  current  in  c.G.s.  units,  n  the  number  of  turns, 
and  I  the  length  of  the  axis  of  the  spiral,  or  if  N  denote  the 
number  of  turns  per  unit  length  of  the  solenoid  then  N=njl 
and  the  value  of  the  force1  is 


2O2.  Astatic  Coil.  If  the  coil  be  as  in  Fig.  195  the 
earth's  magnetic  Held  acts  on  it  and  may  render  it  unsuitable 
for  delicate  observations,  it  can  however  be  rendered  astatic 
by  bending  the  wire  as  in  Fig.  200. 

B 


E 

~~\                        F 

TA 

I 

I 

N 

S 

f                            / 

\     / 

v,               \ 

:            D   H            G 

Fig.  200. 

If  the  current  flow  as  indicated  by  the  arrows  the  face 
of  ABODE  turned  towards  the  reader  is  a  north  pole,  that  of 
EFGH I  a  south  pole,  the  earth  acts  equally  and  oppositely 
on  the  two  and  the  system  is  astatic. 

2O3.  Electromagnetic  Forces.  We  may  regard 
the  above  results  somewhat  differently  thus.  We  have  seen 
that  two  parallel  wires  carrying  currents  in  the  same  direction 
attract.  Lot  us  suppose  the  wires  to  be  perpendicular  to  the 
paper  and  to  cut  it  in  A  and  />',  Pig.  201,  and  let  the  currents 
flow  downwards.  Then  if  the  wires  be  sufficiently  long 
1  If  the  current  is  measured  in  amperes  the  magnetising  force  is 


332 


ELECTRICITY 


[CH.  XX 


compared  with  the  distance  AB,  the  lines  of  magnetic  force 
round  them  are  circles,  as  shewn  by  the  dotted  lines  in  the  figure. 


/  i/"  I  A/\  \  <•  \  \  \ 
\        \  jk.\A>'; \\i*i'j  /   ;    }  ' 


Fig.  201. 

At  a  point  such  as  P  between  A  and  .#  the  magnetic  forces 
due  to  the  two  currents  act  in  opposition,  at  a  point  such  as  Q 
the  directions  of  the  two  forces  coincide.  Thus  to  the  right 
of  B  the  magnetic  field  is  strengthened  by  the  presence  of  -4, 
between  A  and  B  it  is  weakened  ;  the  lines  of  force  due  to  the 
combined  system  will  run  somewhat  as  shewn  in  the  Fig.  201  a. 


Fig.  201  a. 

Now  we  have  seen  in  the  case  of  the  electrostatic  field  that 
many  of  the  phenomena  can  be  explained  by  supposing,  (1)  that 


203] 


ELECTROMAGNETISM 


333 


the  lines  of  force  are  attached  to  the  conductors  in  which  they 
terminate,  and  (2)  that  there  exists  in  the  field  (i)  a  tension 
acting  along  the  lines  of  force  tending  to  shorten  them,  and 
(ii)  a  pressure  at  right  angles  to  the  lines  of  force  tending  to 
press  them  apart.  Clearly  the  result  of  such  action  in  the 
magnetic  field  would  be  to  bring  the  wires  A  and  B  together, 
to  cause  each  to  move  in  a  direction  at  right  angles  to  its  own 
length  and  to  the  lines  of  force  in  the  field. 

We  shall  find  that  the  hypothesis  that  a  force  is  impressed 
on  each  unit  of  length  of  a  wire  carrying  a  current  in  a  mag- 
netic field  which  is  proportional  to  the  product  of  the  strength 
of  the  field  and  the  current,  and  perpendicular  to  both  the  wire 
and  the  lines  of  magnetic  force,  and  acts  in  the  direction  in 
which  the  strength  of  the  field  decreases,  will  enable  us  to 
account  for  the  attractions  and  repulsions  observed,  and  it  is 
a  consequence  of  theory  that  such  a  force  should  exist. 

Thus  consider  a  vertical  wire  in  a  uniform  horizontal 
magnetic  field  H  acting  from  south  to  north.  Let  the  wire 
cut  the  paper  in  Fig.  202,  and  let  the  current  flow  downwards. 


Fig.  202. 

The  lines  of  force  due   to  the  current  i  are  the  dotted 

circles. 


334 


ELECTRICITY 


[CH.  XX 


Let  EO  W  be  the  east  and  west  line  through  0. 

At  a  point  P  between  0  and  E  the  force  due  to  the 
current  is  opposite  to  that  due  to  the  field,  to  the  west  of  0 
the  reverse  is  the  case,  thus  the  field  is  strengthened  by  the 


Fig.  203. 


203] 


ELECTROMAGNETISM 


335 


current  to  the  west  of  0,  weakened  to  the  east.  According  to 
the  law  the  conductor  is  driven  from  west  to  east  by  the 
electromagnetic  force  with  a  force  per  unit  length  equal  to  Hi. 
If  the  current  be  upwards  the  magnetic  force  is  frorn  east  to 
west. 

Fig.  203  gives  a  more  complete  drawing  of  the  same. 

The  magnetic  force  acts  horizontally  from  right  to  left  and 
the  current  runs  vertically  downwards. 

From  these  considerations  we  can  get  a  rule  connecting 
together  the  directions  of  the  current,  the  magnetic  field, 
and  the  electromagnetic  force. 

RULE.  If  a  right-handed  screw  be  placed  with  its  axis 
perpendicular  to  the  plane  containing  the  current  and  the  lines 
of  magnetic  force,  and  turned  so  that  the  direction  of  rotation 
is  from  the  direction  of  the  current  to  that  oj  the  field,  then  the 
point  of  the  screw  moves  in  the  direction  of  the  electromagnetic 
force. 

The  various  directions  are  shewn  in  Fig.  204. 

In  all  cases  the  force  per  unit  length  on  the  conductor  is  equal  to  the 
product  of  the  current  and  the  component 
of  the  field  resolved  at  right  angles  to  the 
current.  If  H  be  the  strength  of  the  field, 
a  the  angle  between  it  and  the  current  i, 
then  the  component  in  question  is  H  sin  a 
and  the  electromagnetic  force  is  Hi  sin  a. 
This  law  was  first  established  by  Ampere 
by  direct  experiment,  the  experiments  how- 
ever are  difficult  and  not  conclusive ;  the 
real  proof  of  the  law  lies  in  the  fact  that 
it  is  made  use  of  to  calculate  the  theoretical 
consequences  of  many  complex  experiments 
and  these  theoretical  consequences  are 
found  to  agree  with  experimental  results. 

Another  form  of  the  rule,  due  to 
Professor  Fleming,  may  sometimes 
be  found  more  convenient.  Extend  Fig>  204. 

the  first  finger  of  the  left  hand  hori- 
zontally, as  in  Fig.  205,  point  the  second  finger  downwards, 
and  the  thumb  to  the  right.     Then   keeping  them  in  these 
relative  positions,  place  the  hand  so  that  the  first  finger  may 
be  parallel  to  the  lines  of  magnetic  force,  and  the  second  finger 


CURRENT 


336 


ELECTRICITY 


[CH.  XX 


parallel  to  the  current,  the  thumb  will  then  give  the  direction 
of  the  electromagnetic  force. 


MAGNETIC  FIELD 


Fig.  205. 

2O4.  Electromagnetic  Action  between  Con- 
ductors. We  may  use  these  results  to  explain  the  attractions 
and  repulsions  described  in  Section  200. 


R      / 

TV 

7P 

A 


M 


Fig.  206. 


Fig.  207. 


203-206]  ELECTROMAGNETISM  337 

For  let  us  suppose  the  wire  LA  M  carrying  a  current  from 
L  to  M  to  be  fixed,  and  consider  another  wire  EAB,  Fig.  206, 
in  which  a  current  flows  from  E  to  B.  Then  the  magnetic 
field  in  which  KA B  is  placed  is  that  due  to  the  current  in  LM. 
At  a  point  such  as  P  the  magnetic  force  is  downwards,  hence 
the  electromagnetic  force  is  along  PR  perpendicular  to  EAB ; 
while  at  Q  it  is  along  QS,  also  perpendicular  to  EAB.  In 
consequence  of  these  forces  EAB  tends  to  set  itself  parallel 
to  LAM. 

Other  cases  may  be  treated  in  the  same  way. 

205.  Electromagnetic     action.       Equilibrium 
condition.     It  follows  as  a  consequence  of  the  above  law 
of  electromagnetic  force  that  a  circuit,  which  is  free  to  move 
in  a  magnetic  field,  when  carrying  a  current,  sets  itself  in  such 
a   position  that  the  number  of  lines  of  magnetic  induction 
which  pass  through  it  from  its  south  face  to  its  north  may 
be  as  large  as  possible. 

A  loop  of  wire  suspended  freely  on  the  ampere  stand, 
Section  199,  sets  itself  .with  its  plane  east  and  west,  and  its 
south  face  pointing  south,  the  axis  of  the  spiral  shewn  in 
Fig.  198  above  will  point  north  and  south. 

If  a  current  be  passed  through  a  perfectly  flexible  conductor 
which  is  supported  in  a  plane  at  right  angles  to  a  uniform 
magnetic  field,  the  conductor  takes  the  form  of  a  circle,  for 
in  this  way  it  includes  in  its  area  the  maximum  number  of 
lines  of  force.  This  experiment  may  be  approximately  realized 
by  the  use  of  a  thin  strip  of  gold-leaf  or  tin-foil  through  which 
a  current  is  passed. 

206.  Motion  of  a  conductor  in  a  magnetic  field. 

The  electromagnetic  force  on  a  conductor  carrying  a  current 
may  be  shewn  in  various  ways. 

Thus  in  Fig.  207,  NS  is  a  strong  cylindrical  magnet  fixed 
in  a  vertical  position.  The  lower  end  D  of  a  wire  CD  is 
supported  so  that  it  can  turn  freely  about  a  vertical,  axis 
through  the  magnet.  The  upper  end  can  turn  about  C,  a 
point  in  the  axis  8ff  produced,  and  the  wire  rests  in  an 
oblique  position.  The  end  D  dips  into  an  annular  mercury 

G.  E.  22 


338 


ELECTRICITY 


[CH.  XX 


cup,  and  from  thence  a  current  can  be  passed  along  the  wire. 
'Lines  of  magnetic  force  issue  from  the  magnet  and  cut  the 
wire.  If  a  current  be  passed  through  the  wire  it  will  be 
set  into  continuous  rotation,  the  direction  of  which  will 
depend  on  that  of  the  current.  If  the  current  be  downwards 
from  C  to  D,  the  end  D  of  the  wire  will  move  from  the 
observer. 

Barlow's  wheel,  Fig.  208,  affords  another  example  of  the 
motion  of  a  conductor  carrying  a  current  in  a  magnetic  field. 


Fig.  208. 

A  star-shaped  disc  can  rotate  easily  about  a  horizontal 
axis.  The  points  of  the  star  as  the  disc  rotates  dip  into  a 
little  mercury  contained  in  a  hollow  in  the  stand  of  the 
instrument,  and  a  current  passes  through  the  disc  between 
the  axis  and  this  mercury. 

A  horseshoe  magnet  is  placed  so  that  its  poles  are  on 
either  side  of  the  mercury  cup ;  thus  the  disc  can  rotate 
between  the  poles  of  the  magnet,  the  lines  of  magnetic  force 
traverse  the  disc  in  a  direction  at  right  angles  to  the  current, 
and  the  result  is  that  the  disc  is  set  into  rotation,  which 
continues  so  long  as  the  current  is  maintained. 

The  rotation  of  a  current  about  a  current  can  be  illustrated 
by  tile  apparatus  shewn  in  Fig.  209.  A  wire  frame  is  pivoted 
on  to  the  top  of  a  pillar  rising  from  the  centre  of  a  circular 
trough  which  contains  copper  sulphate  or  some  conducting 
liquid. 


206-207] 


ELECTROMAGNETISM 


339 


Fig.  209. 


The  trough   is  surrounded   by  a  horizontal   coil  of   wire, 
and  the  connexions  are  such  that  a  current  after  traversing 
this  coil  can  pass  up  the  fixed  stem  and  then  along  the  two 
branches   of    the    wire  £C,    /W,    and 
away  again  through  the  liquid  in  the 
trough. 

The  fixed  coil  produces  a  magnetic 
field  whose  lines  of  force  cut  the  move- 
able  conductors  EG  and  BD.  The 
electromagnetic  forces  on  these  two 
conductors  both  act  at  right  angles  to 
the  paper  but  in  opposite  directions, 
thus  the  moveable  frame  rotates  about 
the  central  pivot. 

207.  Rotation  of  a  magnet 
about  a  current.  If  a  magnet  NS, 
Fig.  210,  be  supported  as  in  the  figure 
so  that  it  can  rotate  about  a  vertical 
wire  in  which  a  current  flows  down- 
wards remaining  always  parallel  to  the 

wire,  the  force  on  the  south  pole  tends  to  bring  it  forward 
from  the  paper,  that  on  the  north  pole  tends  to  depress  it 
below  the  paper,  and  the  movements  of  these  forces  about  the 
axis  being  equal  the  magnet  does  not  rotate. 

But  suppose  the  arrangement  is  as  in  Fig.  211,  where  two 
magnets  are  shewn  pivoted  in  a  mercury  cup  at  A.  At  B  is 
a  wire  dipping  into  an  annular  cup. 

The  current  enters  through  the  central  vertical  rod,  passes 
to  the  annular  cup  by  the  wire  attached  to  the  magnets,  and 
down  the  stand  to  the  binding  screw.  The  north  poles  of  the 
magnets  are  above  the  axial  current,  and  hence  the  force  on 
them  is  less  than  on  the  south  poles  which  are  close  to  the 
central  conductor.  Thus  rotation  takes  place. 

The  results  we  have  just  been  considering  as  to  the  motion 
of  a  circuit  carrying  a  conductor  in  a  magnetic  field  have 
important  practical  applications.  The  action  of  the  electric 
motor  depends  entirely  on  them,  and  Barlow's  wheel  is  in 

22 2 


340 


ELECTRICITY 


[CH.  XX 


a  sense  the  parent  of  the  modern  applications  of  electricity  to 
the  supply  of  motive  power. 


N 


i 


Fig.  210. 


Fig.  211. 


The  principles  of  some  simple  form  of  motors  are  described 
oriefly  in  Section  247. 


CHAPTER   XXI. 


MAGNETISATION  OF   IRON. 

208.  Electromagnets.  The  fact  that  iron  can  be 
magnetised  by  a  current  was  discovered  by  Arago.  Since 
iron  is  magnetised  by  being  brought  into  a  magnetic  field  and 
a  current  flowing  in  a  wire  produces  such  a  field  it  is  clear 
that  the  magnetisation  will  take  place.  Arago  shewed  that 
a  spiral  of  copper  wire  in  which  a  current  is  passing  if  dipped 
into  iron  tilings  became  coated  with  the  filings  like  a  magnet. 
If  again  a  piece  of  copper  wire  covered  with  some  insulating 
material  be  wound  in  a  spiral  on  a  tube  as  in  Fig.  212,  and 


Fig.  2] 2. 

a  current  be  passed  through  the  wire,  then  the  interior  of  the 
tube  becomes  a  field  of  force  and  a  piece  of  steel  placed  in 
the  tube  and  shaken  while  there  becomes  permanently  mag- 
netised. If  the  tube  be  filled  with  a  core  of  soft  iron  the 


342 


ELECTRICITY 


[CH.  XXI 


lines  of  magnetic  force  are  concentrated  through  the  iron 
which  while  the  current  lasts  becomes  a  powerful  magnet; 
on  breaking  the  current  much  of  the  magnetism  of  the  iron 
disappears. 

The  laws  we  have  already  investigated  enable  us  to  tell 
which  end  of  the  iron  core  will  be  a  north  pole.  For  let  the 
current  appear  to  an  observer  looking  at  the  spiral  to  circulate 
in  the  direction  shewn  in  Fig.  213  a.  The  lines  of  force  in 


Fig.  213  a. 


Fig.  213  b. 


the  interior  of  the  spiral  pass  through  the  paper  from  below 
upwards  leaving  the  paper  on  its  upper  side;    thus  the  end 


Fig.  214. 

looked   at  is  a  north  pole  :    if  however  the  direction  of  the 
current  be  as  in  Fig.  213  6,  the  lines  of  force  enter  the  paper 


208-209] 


MAGNETISATION   OF   IRON 


343 


from  above ;  the  end  is  a  south  pole.  This  we  have  seen  already 
in  Section  201.  Electromagnets  take  various  forms  as  shewn 
in  Fig.  214  a,  b  and  c.  In  all  cases  we  have  the  magnetising  coil 
or  coils  wound  on  a  continuous  core  of  soft  iron;  the  core 
should  be  continuous,  for  the  object  is  to  confine  within  the 
iron  as  many  magnetic  lines  as  possible,  and  a  break  in  the 
continuity  permits  the  escape  of  lines  of  magnetic  force  into 
the  surrounding  medium. 


. 


209.  Magnetic  induction.  We  have  already  spoken 
of  magnetisation  produced  by  induction  or  induced  magneti- 
sation :  the  term  magnetic  induction  is  however  used  with  a 
special  significance. 

If  a  piece  of  iron  is  placed  in  a  magnetic  field  a  number  of 
lines  of  force  are  concentrated  into  the  iron  and  after  traversing 
it  pass  out  again  into  the  air. 

At  points  where  the  lines  of  force  enter  the  iron  we  have 
a  series  of  south  poles,  at  points  when  they  leave  we  have  north 
poles,  in  each  case  formed  by  induction. 

The  force  both  throughout  the  iron  and  in  the  surrounding 
air  depends  in  part  on  this  induced  magnetism,  and  this  again 
depends  on  the  strength  of  the  original  field  and  on  the  shape 
and  magnetic  quality  of  the  iron. 

Again,  if  we  suppose  a  cavity  formed  in  the  iron,  Fig.  215, 
the  force  on  a  unit  north  magnetic 
pole  placed  in  the  cavity  will  depend 
in  part  upon  the  original  strength  of 
the  field,  in  part  on  the  shape  of  the 
cavity. 

Now  let  //  be  the  strength  of  the 
field  and  let  us  consider  a  portion  of 
the  surface  of  the  iron  where  the 


Fig.  215. 


lines  of  force  of  the  field  enter  it  at  right  angles.  South  polar 
or  negative  magnetism  will  be  induced  on  this  area,  and  the 
amount  of  this  will  depend  on  H  and  on  the  iron.  Let  us 
suppose  the  quantity  of  south  polar  magnetism  induced  per 
unit  of  area  to  be  /  and  let  us  put  /  =  KJEf,  then  K  is  called 


344 


ELECTRICITY 


CH.  XXI 


the  coefficient  of  induced  magnetisation.  Experiment  shews 
that  K  is  not  independent  of  the  strength  of  the  magnetising 
field;  it  depends  on  this  and  on  the  past  history  of  the  iron. 

21O.     Magnetic   force   within   a  mass   of   iron. 

Now  let  us  take  a  case  in  which  the  field  is  uniform  in  the 
iron  and  imagine  a  long  narrow  cylindrical  cavity  as  in  Fig.  216, 
cut  in  the  iron  with  its  axis  in  the  direction  of  the  magnetising 
force  and  its  ends  at  right  angles  to  that  direction. 


Fig.  216. 


Fig.  217. 


On  one  end  of  this  cavity  where  the  lines  of  force  leave  it 
we  shall  have  an  amount  of  induced  magnetism  /  or  Kff  per 
unit  of  area,  on  the  other  end  there  will  be  —  K!£  per  unit  of 
area.  Thus  if  the  ends  be  a  square  centimetres  in  area  the 
quantities  of  magnetism  on  them  will  be  la  and  —la  respectively. 
Since  the  curved  walls  of  the  cavity  are  parallel  to  the  lines 
of  force  of  the  field  no  lines  of  force  either  enter  or  leave 
these,  thus  no  induced  magnetism  is  on  them.  Thus  the  force 
within  the  cavity  is  the  original  magnetising  force  H  together 
with  the  forces  arising  from  the  quantities  of  magnetism 
la  and  -  la  on  the  ends.  Now  if  the  cavity  be  very  narrow 
a  is  very  small,  and  the  quantities  of  magnetism  concerned 
will  be  very  small,  while  if  the  cavity  be  very  long  they  will 
be  at  a  great  distance  from  the  point  at  which  the  force  is 
being  estimated,  thus  the  amount  they  contribute  to  the  force 
will  be  vanishingly  small.  Hence  the  resultant  magnetic 
force  at  a  point  within  the  cavity  will  be  H,  the  magnetising 
force. 

211.     Magnetic  force  in  a  crevasse.     Now  let  us 

suppose  that  the  cavity  is  extremely  short,  a  kind  of  narrow 


209-211]  MAGNETISATION   OF    IRON  345 

crevasse,  cut  across  the  lines  of  force  as  in  Fig.  217.  Then 
the  two  ends  with  the  magnetic  charges  a/  and  —a/  will 
be  like  two  small  oppositely  charged  flat  plates  close  together 
charged  to  surface  densities  /  and  —  /  respectively.  They 
resemble  the  plates  of  an  electric  condenser. 

Now  we  have  already  seen,  Section  43,  that  if  a-  be 
the  surface  density  of  a  condenser  with  air  for  its  dielectric 
the  force  between  the  plates  is  47ro-,  and  that  it  acts  from  the 
positive  to  the  negative  plate.  But  magnetic  forces  follow 
the  inverse  square  law  in  the  same  way  as  electrical.  Hence 
we  can  infer  by  exactly  similar  reasoning  that  the  force  in  a 
crevasse  cavity  cut  normal  to  the  magnetising  force  due  to 
magnetism  with  density  /  and  —  /  on  its  flat  faces  is  4?r/  acting 
from  the  positive  to  the  negative  face. 

In  order  to  get  the  actual  force  in  the  crevasse  we  must 
add  to  this  force  47r/  the  magnetising  force  H  ;  and  we  have 
as  the  result  H  +  4-Tr/,  or  putting  for  7  its  value  K//  we  get 
as  the  force  H+  l-mtH  or  H  (I  + 


This  quantity,  the  resultant  magnetic  force  in  the  crevasse 
cavity,  is  known  as  the  Magnetic  Induction,  and  is  usually 
denoted  by  B.  Thus  if  we  write  1  -f  4™  =  ^  we  have 


The  quantity  /x  is  known  as  the  permeability. 
We  are  thus  led  to  the  following  definitions. 

DEFINITION.  The  ratio  of  the  quantity  of  magnetism 
induced  per  unit  of  area  on  a  surface  normal  to  the  direction 
of  the  magnetising  force  to  that  force  is  known  an  the 
Coefficient  of  induced  magnetisation. 

It  is  denoted  by  K,  and  we  have  /=  xH. 

DEFINITION.  The  resultant  magnetic  force  within  a  narrow 
crevasse  cut  in  a  magnetic  medium  and  bounded  by  faces 
perpendicular  to  the  magnetising  force  is  known  as  the 
Magnetic  Induction. 

It  is  usually  denoted  by  B,  and  we  have 

+47T/C. 


346  ELECTRICITY  [CH.  XXI 

DEFINITION.  The  ratio  of  the  magnetic  induction  to  the 
'magnetising  force  is  known  as  the  Permeability. 

It  is  usually  denoted  by  p. 
Hence  B  =  pH. 

212.  Magnetic  permeability.  We  may  obtain  these 
results  somewhat  differently  thus. 

Consider  a  bar  magnet,  uniformly  magnetised,  let  /  be  the 
intensity  of  its  magnetisation,  a  the  area  of  its  cross-section, 
then  the  strength  of  its  north  pole  is  /a.  Now  from  the 
definition  it  can  be  shewn  that  from  a  pole  of  strength  ra 
the  number  of  lines  of  force  which  issue  is  kirin.  Again,  the 
magnetic  force  at  any  point  is  the  number  of  lines  of  force 
which  cross  unit  area  placed  at  right  angles  to  the  force  in 
such  a  position  as  to  contain  the  point. 

Now  imagine  the  magnet  bent  round  into  the  form  of  a 
ring  so  that  its  north  and  south  poles  may  almost  coincide. 
It  will  produce  no  external  magnetic  field  except  just  in  the 
gap  between  the  poles,  and  the  number  of  lines  of  force  which 
traverse  the  gap  is  iirla.  Hence  the  number  of  lines  of  force 
per  unit  area  in  the  gap  is  47r7,  and  this  is  the  magnetic  force 
in  the  gap. 

If  the  magnetisation  of  the  magnet  be  produced  by 
induction  we  must  add  to  this  force  H  the  magnetising  force, 
and  we  get  as  the  resultant  force  in  the  gap  H  +  kirl. 

If  this  force  be  defined  as  the  induction  and  denoted  by  B 
then 

B  =  H  +  47T/. 

But  7  =  KJFf. 

Hence  B  =  H+  ^KH 

=  A(l+4:TTK)H 


Thus  the  magnetic  induction  in  any  medium  is  the  number 
of  lines  of  force  which  cross,  per  unit  area  a  narrow  gap  in  the 
medium,  the  surfaces  of  the  gap  being  at  right  angles  to  the 
lines  of  force. 


211^214]  MAGNETISATION   OF    IRON  347 

It  is  clear  from  the  above  that  //,  is  unity  and  the  measure 
of  the  induction  the  same  as  that  of  the  force  for  all  media 
for  which  K  vanishes,  that  is  to  say  for  all  except  magnetic 
media.  Experiment  shews  that  /c  is  excessively  small, 
practically  zero,  except  in  the  case  of  iron,  nickel,  and  cobalt. 
The  value  of  /z  depends  on  the  magnetising  force.  For  iron  in 
moderate  fields  it  may  range  between  400  and  2500.  for  nickel 
and  for  cobalt  its  maximum  limit  is  about  200. 

In  certain  media  K  is  a  small  negative  quantity  ;  when  this 
is  the  case  fj.  is  less  than  unity;  such  media  are  called 
diamagnetic.  Iron  and  media  for  which  K  is  positive  and 
JJL  greater  than  unity  are  called  paramagnetic. 

In  diamagnetic  media  the  induced  magnetism  is  opposite 
to  the  magnetic  force. 

213.  The  magnetic  circuit.  Now  we  know  from 
Ohm's  law  that  if  C  be  the  current  of  electricity  which  crosses 
unit  area  of  a  conductor  whose  conductivity  is  k  and  in  which 
the  electric  force  is  JS,  then 

C  =  kE. 
We  may  compare  this  with  the  equation 


connecting  together  the  induction  or  flow  of  magnetic  lines 
of  force  per  unit  area,  the  permeability  and  the  magnetising 
force,  the  two  are  clearly  analogous.  If  we  treat  the  magnetic 
induction  as  a  flux  or  current  per  unit  area  it  is  related  to  the 
permeability,  and  the  force,  in  the  same  way  as  the  current 
per  unit  area  is  related  to  the  conductivity  and  the  electric 
force.  We  may  speak  of  the  magnetic  circuit  in  the  same 
way  as  we  use  the  expression  the  electric  circuit.  There  is, 
however,  this  great  difference  that  k,  the  conductivity,  does 
not  depend  on  JE,  the  electric  force,  while  /x,  the  magnetic 
permeability,  does  depend  on  H.  Still  the  analogy  is  often 
useful. 

214.  Magnetic  Reluctance.  Again,  let  us  consider 
a  magnetic  circuit  of  uniform  section  a  and  let  it  be  subject 
to  a  uniform  force  H.  Let  I  be  the  length  of  the  circuit  and 


348  ELECTRICITY  [CH.  XXI 

Oj  ,  fi2  the  magnetic  potentials  at  its  two  ends,  then  since  the 
force  is  the  space-rate  of  change  of  potential,  we  have 


Let  B  be  the  total  flow  of  induction  through  the  circuit. 
Then  B  =  Ba. 

Hence  total  flow  of  induction 

=  B  =  Ba.  —  a.jjiH 


But  if  C  be  the  total  flow  of  electricity  in  a  circuit  of 
section  a,  conductivity  k,  and  length  I,  Vlt  F2  being  the  electric 
potentials  at  the  ends, 

then  C'=y(F1-ra), 

while  l/ka  is  the  resistance  of  the  conductor. 

Thus  if  we  define  ///xa  as  the  magnetic  resistance  or  re- 
luctance of  the  circuit,  we  can  apply  the  laws  governing  the 
flow  of  electricity  to  magnetism.  For  example,  if  we  suppose 
a  body  is  composed  of  a  series  of  portions  of  lengths  llt  12,  ... 
sections  04,  03,  ...  and  permeabilities  ftj,  ^.2,  ...,  then  the  total 
reluctance  is 


and  the  total  flow  of  induction,  if  O15  O2  are  the  potentials  at 
the  beginning  and  end  of  the  circuit,  is 


Thus  it  is  clear  that  if  we  wish  to  produce,  for  a  given 
difference  of  magnetic  potential,  a  large  flow  of  induction  we 
must  keep  the  reluctance  small. 

Now  we  have  already  stated  that  for  a  given  magnetising 
force  the  permeability  of  iron  is  enormously  greater  than  that 
of  any  other  medium,  so  that  to  secure  large  inductions  we 
must  use  iron  and  keep  the  air  gaps  in  our  circuit  as  narrow 
as  possible,  that  is,  keep  values  of  I  which  correspond  to 
the  small  values  of  /*  very  small. 


214-215]  MAGNETISATION    OF    IRON  349 

Thus  \ve  see  how  it  is  that  by  putting  an  iron  core  into  a 
spiral  carrying  a  current  we  increase  enormously  the  magnetic 
force  at  the  ends  of  the  spiral,  the  magnetic  reluctance  of  the 
circuit  is  reduced  and  the  flow  of  induction  increased. 

215.     Measurement   of   magnetic  permeability. 

Several  methods  have  been  devised  for  the  measurement  of 
K  and  /A.  Some  of  these  can  be  best  described  after  we  have 
considered  the  phenomena  of  electromagnetic  induction.  In 
the  magnetometric  method  the  arrangements  described  in 
Section  92  for  the  determination  of  M/ff  are  employed. 

The  iron  to  be  magnetised  takes  the  form  of  a  thin  rod  and 
is  placed  inside  a  magnetising  spiral.  When  a  current  passes 
it  becomes  a  magnet  and  will  deflect  a  small  magnet  placed 
near.  From  this  deflexion  the  magnetic  moment  induced  by 
the  current  can  be  found  in  terms  of  the  inducing  force  ;  but 
this  force  is  known  if  the  current  be  known  and  hence  K  and 
p,  can  be  determined. 

The  experiment  usually  is  conducted  thus. 

EXPERIMENT  53.  To  determine  by  means  of  the  magnetometric 
method  the  coefficient  of  induced  magnetisation,  or  susceptibility, 
and  the  permeability  of  an  iron  rod. 

The  iron  rod  NS,  Fig.  218,  which  should  be  about  40  cm. 
long  and  1  to  2  millimetres  in  diameter  is  placed  inside  a  mag- 
netising spiral  consisting  of  a  thin  coil  some  50  centimetres  in 
length  wound  with  one  or  two  layers  of  insulated  wire.  The 
iron  rod  lies  along  the  axis  of  the  coil  and  this  is  directed  east 
and  west. 

A  small  magnetometer,  usually  one  with  a  mirror  magnet, 
is  placed  at  a  point  on  the  axis  produced  distant  r  centimetres 
from  the  centre  of  the  rod. 

On  passing  a  current  through  the  coil  the  iron  rod  becomes 
a  magnet  and  the  magnetometer  needle  is  deflected.  Let  <£  be 
the  angle  of  deflexion,  M  the  magnetic  moment  of  the  induced 
magnetisation,  21  the  length,  and  A  the  diameter  of  the  rod, 
which  we  suppose  to  be  circular  in  section. 

Then  if  F  be  the  strength  of  the  field  at  the  magnetometer 
due  to  the  magnetised  iron,  H  the  strength  of  the  earth's  field, 


350  ELECTRICITY  [CH.  XXI 

we  know  that,  since  the  directions  of  F  and  H  are  at  right- 
angles, 

F=Htan<]>. 

Again,  the  strength  of  each  pole  of  the  iron  rod  is  M/21, 
one  pole  is  at  a  distance  r  -  I,  the  other  at  a  distance  r  +  I  from 
the  magnetometer,  so  that  if  we  suppose  F  is  due  entirely  to 
the  magnetism  of  the  rod,  we  have 


Hence  we  have 


But  M  is  due  to  the  magnetism  induced  in  the  rod  by  the 
current  in  the  spiral.  Let  X  be  the  magnetic  force  at  any 
point  in  the  axis  of  the  spiral  due  to  this  current  and  let  /  be 
the  induced  magnetisation.  We  assume  this  force  to  be  the 
same  along  that  part  of  the  axis  which  is  occupied  by  the  rod. 
Then,  if  K  is  the  coefficient  of  induced  magnetisation  or  the 
susceptibility,  we  have 

I=KX. 

Since  the  area  of  the  end  of  the  rod  is  TTO?  the  strength  of 
each  pole  is  TTO?!  and  M  the  magnetic  moment  of  the  rod 
is  therefore  l-TraHI  or  lircflKX. 

Substituting  this  value  for  M  we  have 

(r>-P?H^ 
K  =  -  -  zy  -=.  tan  <£, 
kirraH  X 

but  if  N  be  the  number  of  turns  per  unit  length  of  the  spiral 
then  since  X  is  due  to  the  current  i  amperes  we  have  (§  201) 


Now  i  can  be  measured  by  an  ammeter  or  in  some  other 
convenient  way,  thus  X  is  known  and  substituting  for  it 
we  have  finally 

2-22# 

"XT-  tail   9- 

Nl 


One  or  two  points  require  to  be  noticed. 


215]  MAGNETISATION    OF    IRON  351 

In  the  first  place  the  field  F  is  not  due  solely  to  the  action 
of  the  iron  rod ;  the  current  in  the  spiral  would,  even  if  the 
rod  were  removed,  produce  an  effect  on  the  magnetometer. 
This  can  be  measured  or  allowed  for  by  making  an  experiment 
without  the  rod  in  position ;  the  deflexion  observed  will  then 
be  due  to  the  current  only  ;  it  is  better  however  to  compensate 
for  it  by  inserting  in  the  current  circuit  a  few  turns  of  wire 
near  the  magnetometer.  These  can  easily  be  adjusted  so  that 
the  effect  they  produce  on  the  magnet  .shall  be  equal  and 
opposite  to  that  of  the  spiral.  To  do  this  the  iron  is  removed 
and  the  position  of  the  additional  turns  altered  until  no  effect 
is  observed  on  the  magnetometer  whatever  be  the  current  in 
the  circuit.  Then  when  the  iron  is  in  position  the  whole 
effect  is  due  to  its  induced  magnetisation.  Again,  it  has 
been  assumed  that  the  magnetising  field  is  solely  that  due 
to  the  current ;  but  when  the  iron  becomes  magnetised  the 
field  in  its  interior  is  in  part  due  to  its  own  magnetisation 
and  thus  part  of  the  field  is  opposite  in  direction  to  the 
magnetising  field,  for  if  the  latter  act  from  S  to  N  (Fig.  218) 
then  S  becomes  a  south  pole,  N  a  north,  and  the  field  in  the 
magnet  due  to  these  poles  is  from  N  to  S. 


Fig.  218. 

Thus  the  magnetising  field  is  not  strictly  that  due  to  the 
current,  as  has  been  assumed.  The  difficulty  is  met  to  some 
extent  by  making  the  rod  long  and  thin. 

The  arrangement  of  apparatus  is  shewn  in  Fig.  218,  in 
which  NS  is  the  rod,  CD  the  magnetising  spiral,  EF  the 
compensating  coil,  M  the  magnetometer,  R  is  a  resistance  for 


352 


ELECTRICITY 


[CH.  XXI 


regulating  the  current,  G  the  ammeter,  and  K  a  key.  preferably 
a  reversing  switch,  for  making  and  breaking  the  circuit. 

216.  Curve  of  magnetic  induction.  When  the 
value  of  K  has  been  found  thus,  the  value  of  p.  can  be 
calculated  by  the  formula  /x  -  1  +  4?™  and  B  is  given  by 
multiplying  //,  by  the  magnetising  force  X. 

A  series  of  experiments  can  be  made  by  gradually  reducing 


12000 
11000 
10000 

eooo 

8000 
700O 
6000 
5000 
4000 
3000 
2000 

^ 

^ 

/ 

/ 

^^ 

/ 

^ 

/ 

/ 

0 
d 
CO 

/ 

CJ 

^ 

/ 

o 
i^ 
kj 

/ 

•  ^: 

CO 

1 

1 

/ 

I 

J 

MAGNETIC  FORCE 

i         i         i 

123456789          K 

Fig.  219. 

the  resistance  in  circuit  and  thus  increasing  the  magnetising 
force ;  if  this  be  done  the  results  may  be  plotted  as  a  curve  in 


215-216] 


MAGNETISATION   OF   IRON 


353 


which  the  ordinates  represent  B  and  the  abscissae  the  values 
of  X,  the  magnetising  force;  if  the  iron  be  initially  unmag- 
netised,  and  this  must  be  tested  for  in  the  usual  way,  the  curve 
will  have  the  form  given  in  Fig.  219. 

If  at  any  stage  the  current  be  broken  the  iron  will 
remain  magnetised,  but  the  amount  of  this  permanent 
magnetisation  will  depend  largely  on  the  method  adopted 
for  breaking  the  current,  still  the  residual  magnetic  moment 


1200O 

11000 
10000 
9000 
8OOO 
7000 
6000 
5000 
4000 

^ 

d 

o 
^ 

—  —  i 

ealdua 

^-  

. 

ca 

^ 

X 

^^ 

h  S£ 

K: 

tej 

_^ 

/ 

Q3 

•*c 

^ 

/ 

1 

/ 

1000 
0 

1 

^ 

MAGNETIC  FORCE 

Fig.  220. 

and  the  residual  induction  can  be  calculated,  and  if  this  be 
done  for  various  values  of  X  a  curve  of  residual  magnetisation 
can  be  found  as  in  Fig.  220. 

G.  E.  23 


354 


ELECTRICITY 


[CH.  XXI 


217.  Hysteresis.  The  behaviour  of  iron  in  a  magnetic 
field  can  be  more  completely  investigated  by  carrying  the 
current  through  a  complete  cycle.  Starting  from  zero  current 
the  resistance  in  the  circuit  is  gradually  diminished,  thus 
increasing  the  current  until  the  maximum  value  desired  is 
reached.  The  current  is  then  reduced  to  zero,  after  which 
its  direction  is  reversed  and  it  is  carried  on  by  similar  stages 
until  the  former  maximum — only  with  the  direction  of  flow 
changed — is  reached.  From  this  point  it  is  carried  back 
through  zero  to  the  first  positive  maximum. 

When  this  is  done  it  is  found  that  the  curve  for  11  has  the 
form  shewn  in  Fig.  221. 


Fig.  221. 

Starting  from  0  with  the  iron  demagnetised,  it  rises 
gradually  at  first,  then  more  steeply,  and  after  becoming 
very  steep  the  slope  gradually  falls  until  the  line  is  almost 
parallel  to  the  horizontal  axis.  At  A  the  maximum  force 
and  the  maximum  induction  are  reached. 

The  force  is  now  reduced  and  the  induction  falls,  but  it 
does  not  return  along  the  line  APO  but  along  a  line  ACB', 
lying  above  it  and  to  the  left,  cutting  the  vertical  axis  in  (7 
and  the  horizontal  in  B' , 


217-218]  MAGNETISATION   OF   IRON  355 

The  induction  for  each  value  of  the  force  is  greater  than 
the  value  it  had  for  the  same  force  on  the  outward  journey. 
Thus  let  QPN  parallel  to  OY  meet  OX  in  N,  then  when  the 
iron  was  first  being  magnetised,  for  the  force  ON  the  induction 
was  PN;  when  the  magnetisation  is  being  reduced  from  that 
corresponding  to  A,  for  the  same  force  O^Tthe  induction  is  QN, 
which  is  greater  than  PN  and  corresponds  on  the  outward 
curve  to  a  force  greater  than  ON. 

The  induction  lags  behind  the  magnetising  force;  when  the 
force  has  reached  zero  the  induction  is  CO  and  the  force  has 
to  be  made  negative  and  equal  to  OB'  before  the  induction 
becomes  zero.  As  the  force  increases  still  further  negatively 
the  induction  becomes  negative  until  when  the  negative 
maximum  is  reached  for  the  force  the  point  A'  of  the  diagram 
represents  the  condition  of  the  iron.  As  the  force  again 
returns  to  zero  and  after  passing  it  approaches  its  former 
maximum  at  A,  the  curve  A'C'BA  is  traced  by  the  induction 
and  this  curve  is  found  to  be  symmetrical  with  respect  to  the 
axes  with  the  position  ACB'A'. 

The  name  hysteresis — a  lagging  behind — has  been  given  to  this 
phenomenon  because  in  all  cases  the  induction  for  a  given  value  of  the 
force  lags  behind  the  value  it  would  have  for  that  same  value  of  the  force 
if  the  original  curve  of  magnetisation  from  the  demagnetised  condition 
had  been  followed. 

We  might  have  drawn  a  similar  curve  for  the  induced  magnetisation 
and  drawn  similar  conclusions  from  its  form. 

218.  Theories  of  Magnetisation.  Ewing's  Model, 
Section  88,  enables  us  to  understand  some  of  the  changes 
which  probably  go  on  in  a  piece  of  iron  which  is  being  mag- 
netised, and  indeed,  it  has  been  shewn  by  direct  experiment, 
that  an  assemblage  of  a  large  number  of  small  compass  needles 
behave,  when  subject  to  a  gradually  increasing  magnetic  field, 
exactly  in  the  same  way  as  the  bar  or  rod  of  iron. 

At  first  the  magnets  are  arranged  in  closed  circuits  ;  the 
assemblage  produces  no  external  field.  When  a  small  mag- 
netic force  is  applied  some  few  of  the  needles  are  disturbed, 
and  on  the  whole,  there  is  a  tendency  for  the  needles  to  set 
themselves  parallel  to  the  field ;  but  the  mutual  forces  between 
the  needles  restrain  this  tendency,  and  the  magnetic  moment 

23—2 


356  ELECTRICITY  [CH.  XXI 

of  the  bar  remains  small.  As  the  force  is  increased,  it 
gradually  overcomes  the  mutual  attractions ;  more  and  more 
of  the  circuits  are  broken  up,  and  for  certain  values  of  the 
external  field  depending  on  the  strength  of  the  mutual  action 
between  the  compass  needles,  this  destructive  action  goes  on 
very  rapidly,  in  other  words  the  induction  increases  very 
rapidly  with  the  force,  we  are  on  the  steep  part  of  the  curve. 
For  still  larger  values  of  the  force  nearly  all  the  circuits  have 
been  broken,  the  increase  of  induction  becomes  less  rapid 
until  a  condition  is  reached  when  there  are  no  groups  left,  the 
axes  of  all  the  magnets  are  parallel  to  the  field,  the  iron  is 
magnetised  to  saturation. 

Again,  it  is  clear  that  the  configuration  of  the  magnets, 
corresponding  to  any  given  value  of  the  force,  depends  in  part 
upon  the  force,  in  part  upon  the  configuration  when  the  force 
was  applied. 

If  we  imagine  the  assemblage  demagnetised  when  the  force 
is  applied,  we  shall  get  one  result  depending  on  the  balance 
between  the  impressed  force  and  the  mutual  forces  between 
the  magnets  ;  if  we  imagine  the  assemblage  saturated  and  the 
external  field  to  be  then  reduced  until  it  has  the  same  value 
as  in  the  previous  case,  the  configuration  will  usually  be 
different,  because  the  mutual  forces  which  have  been  overcome 
are  different. 

The  configuration  of  the  magnets  at  any  moment  depends 
upon  the  external  field,  and  upon  the  configuration  when  the 
field  was  applied.  In  this  way  the  model  enables  us  to  explain 
the  phenomenon  of  hysteresis. 

219.  Energy  needed  to  Magnetise  Iron.  Work 
must  be  done  to  magnetise  a  piece  of  iron,  and  this  for 
two  reasons  at  least,  (1)  a  magnetic  field  is  produced  and  the 
magnetised  iron  can  do  work,  and  (2)  energy  is  necessary  to 
break  up  the  closed  molecular  circuits  in  the  iron,  to  move  the 
molecules  against  their  mutual  attractions. 

By  the  demagnetisation  of  the  iron  the  first  part  of  the 
energy  can  be  recovered.  Not  so  however  with  the  latter 
portion.  As  the  molecular  chains  break  up  the  molecules  are 
thrown  into  a  state  of  disturbance,  oscillating  backwards  and 
forwards,  and  part  of  the  energy  supplied  is  frittered  away  as 


218-219] 


MAGNETISATION    OF   IRON 


357 


heat.  This  part  of  the  energy  is  irrecoverable  as  magnetic 
energy,  the  process  is  irreversible ;  by  carrying  the  iron 
through  a  complete  cycle  an  amount  of  heat  is  produced  which 
it  can  be  shewn  is  measured  per  unit  of  volume  by  the  area  of 
the  hysteresis  loop  divided  by  47r. 

We  can  obtain  a  connexion  between  the  area  of  the  loop 
and  the  work  done  in  magnetising  the  iron,  thus : 

Consider  a  small  magnet,  let  in  and  -  m  be  the  strengths 
of  its  two  poles,  H  the  field  in  its  neighbourhood,  and  let  us 
consider  the  work  required  to  increase  the  pole  strengths  by 
small  amounts  m  and  -  m  respectively.  Work  will  be  done 
in  carrying  the  amount  —  in  to  the  south  pole  from  outside 
the  field,  and  also  in  carrying  -i-  m  to  the  north  pole.  As  we 
have  seen,  when  discussing  the  theory  of  the  potential,  the 
work  will  not  depend  on  the  path  followed.  Now  we  may 
imagine  in  brought  to  S  by  exactly  the  same  path  as  was 
travelled  by  -  m  and  then  carried  directly  from  S  to  N.  The 
amounts  of  work  done  in  carrying  m!  and  —  m  to  S  will  be  equal 
and  opposite,  the  amount  of  work  done  in  carrying  m  from  # 
to  N  is  %m'Hl,  where  21  is  the  length  of  the  magnet  Stf. 

Now  the  poles  are  increased 
in  strength  by  increasing  the 
induced  magnetisation  /  and  if 
this  be  increased  by  a  small 
amount  S/  and  if  a  be  the  area 
of  a  section  of  tjie  magnet  nor- 
mal to  the  axis,  then  m  =  aS7. 

Hence  the  work  done 

=  2/0  mi. 

But  2la  is  the  volume  of  the 
magnet,  hence  the  work  done 
per  unit  of  volume  is  H8I. 

Hence  looking  upon  the  final 
state  as  having  been  reached  by 
successive  additions  to  the  mag- 
netisation, we  see  that  the  work 
done  may  be  written  3/787  where 


Fig.  222. 
is  an  abbreviation  for 


the  sum  of  quantities  such  as  the  above. 


358 


ELECTRICITY 


[CH.  XXI 


But  if  we  draw  a  curve  OP,  Fig.  222,  such  that  the 
abscissa  of  a  point  P  represents  the  force  H  and  the 
ordiuate  the  magnetisation  7,  then  as  in  Section  37  we  can 
shew  that  the  area  0PM,  where  PM  is  parallel  to  the  line 
of  force,  is  equal  to  H8I.  Hence  the  area  0PM  measures  the 
energy  per  unit  volume  required  to  magnetise  the  iron. 

Now  in  Fig.  223  let  A'C'BA  be  the  curve  of  magnetisation 
starting  from  the  point  A'  at  which  the  force  and  magnetisation 
have  their  maximum  negative  values,  AGB'A'  the  curve  of 
demagnetisation  and  let  AM,  A'M'  be  parallel  to  OX. 


Fig.  223. 

Then  in  passing  from  C"  to  A  an  amount  of  energy  C'AM  is 
absorbed  per  unit  volume,  but  in  passing  from  A  to  C  an  amount 
CAM  is  returned.  Thus  an  amount  GAG'  is  absorbed  in 
passing  by  the  curve  from  C"  to  A  and  then  to  G.  Similarly 
the  amount  absorbed  in  passing  from  C  to  A'  and  then  to  C" 
i&'G'A'C.  Hence  the  amount  absorbed  in  going  round  the 


219] 


MAGNETISATION    OF    IRON 


359 


cycle  is  represented  by  the  area  of  the  loop.  It  should  be 
noted  that  in  this  curve  the  ordinates  represent  the  induced 
magnetisation  /,  not  the  induction  B  as  in  Fig.  221,  and  the 
energy  absorbed  is  equal  to  the  area  of  the  magnetisation  loop. 

We  obtain  the  induction  curve  from  this  one  thus  : 

Since  we  have  B  —  H+  4:irl, 

we  can  draw  the  magnetisation  curve. 

Let  P'PN,  Fig.  224,  be  parallel  to  OY  and  let  Q'Q  be  the 
points  on  the  induction  curve  corresponding  to  P  and  P. 


Fig.  224. 


Then  since 
we  have 


Q'N  =  ON 


Hence  QQ'  =  4irPP'. 

Thus  if  we  draw  consecutive  ordinates  it  is  clear  that  the 
areas  of  corresponding  strips  of  the  curves  are  as  1  to  4?r,  for 
their  breadths  are  equal  and  their  heights  are  as  QQ'  to  PP. 


360  ELECTRICITY  [CH.  XXI 

Tims  the  area  of  the  induction  loop  is  4?r  times  that  of  the 
magnetisation  loop. 

Hence  the  area  of  the  hysteresis  curve  as  denned  in 
Section  217  is  4?r  times  the  energy  per  unit  volume  required 
to  carry  the  iron  round  the  cycle. 

When  the  area  of  the  loop  is  small  we  infer  that  the 
molecules  follow  the  changes  of  the  magnetising  force  readily ; 
all  the  energy  spent  in  magnetisation  can  be  obtained  by  de- 
magnetising the  iron.  When  on  the  other  hand  the  loop  is 
large,  an  additional  amount  of  energy  is  required  to  magnetise 
the  iron  and  this  is  not  returned  when  the  iron  is  demagnetised. 

The  iron  is  heated  and  the  rise  of  temperature  is  a  measure 
of  this  hysteresis  loss,  which  can  be  calculated  by  measuring 
the  area  of  the  loop  and  the  volume  of  the  iron. 


CHAPTER  XXII. 


ELECTROMAGNETIC    INSTRUMENTS. 

2 2O.  Moving  Coil  Galvanometers.  The  mutual 
action  between  a  wire  carrying  a  current  and  a  magnetic 
field,  or  between  a  similar  wire  and  a  piece  of  soft  iron,  is 
made  use  of  in  many  forms  of  instrument  for  the  measure- 
ment of  current  or  of  electromotive  force. 

We  have  already  seen  how  to  utilize  it  in  the  ordinary 
galvanometer  in  which  the  current  circulates  in  a  fixed  coil 
and  a  magnet  is  delicately  suspended  near  its  centre.  In 
some  more  modern  forms  of  instruments — moving  coil  galva- 
nometers— the  magnet  is  fixed,  and  the  coil  carrying  the 
current  moves. 

Such  an  instrument  is  shewn  in  Fig.  225.  The  magnet  is 
in  the  form  of  a  powerful  horseshoe  magnet,  the  poles  of 
which  are  brought  near  together.  The  coil  of  wire  liarigs  in 
the  field  between  the  poles,  its  plane  being  parallel  to  the 
lines  of  force,  and  the  axis  therefore  of  the  magnet,  to  which 
it  is  equivalent  when  a  current  is  passed  through  it,  is  at  right 
angles  to  the  lines  of  force.  When  a  current  circulates  in 
the  coil  it  tends  to  turn  so  as  to  include  in  its  area  the 
maximum  number  of  lines  of  force ;  if  the  coil  were  suspended 
quite  freely  it  would  turn  through  a  right  angle,  and  set 
itself  perpendicularly  to  its  former  position.  But  the  wires 
suspending  the  coil  and  by  means  of  which  the  current  is 
brought  to  it,  exercise  a  constraint  on  the  coil,  and  in 
consequence  it  only  turns  until  the  couple  due  to  the  electro- 
magnetic action  balances  that  due  to  the  constraint.  In  some 


362 


ELECTRICITY 


[CH.  XXII 


instruments  the  coil  hangs  by  a  single  wire,  a  second  wire 
is  continued  downwards  below  the  coil,  being  stretched  fairly 
tight.  The  coil  in  turning  twists  this  wire  and  the  constrain- 
ing couple  arises  from  the  torsion  thus  produced.  The  current 


Fig.  225. 

passes  in  from  above  and  out  below  the  coil.  In  very  sensitive 
instruments  the  lower  wire  is  loose,  the  couple  arises  from  the 
torsion  of  the  upper  wire  which  is  stretched  by  the  weight  of 
the  coil. 

In  other  instruments  again  the  suspension  is  bifilar ;  the 
coil  hangs  from  two  very  fine  parallel  wires,  the  current  enters 
by  one  and  leaves  by  the  other ;  as  the  coil  turns  it  is  lifted 
slightly  by  the  action  of  the  wires  and  its  weight  supplies  the 
restraining  couple. 

The  motion  of  the  coil  may  be  indicated  by  a  pointer 
moving  over  a  scale,  or  by  a  ray  of  light  reflected  from  a 
mirror  attached  to  the  coil. 

The  relation  between  the  strength  of  the  current  and  the 
deflexion  will  depend  on  the  distribution  of  the  lines  of 
magnetic  force  in  the  field.  By  properly  choosing  the  form 
of  the  magnet  this  can  be  arranged  so  that  the  deflexion  is 
very  accurately  proportional  to  the  current;  in  such  an 
instrument  the  current  causing  a  given  deflexion  is  read  off 
directly. 


220-221]          ELECTROMAGNETIC    INSTRUMENTS 


363 


In  another  form  of  the  arrangement,  the  coil  is  mounted 
on  pivots  between  two  jewels,  and  carries  a  pointer  which 
moves  over  a  uniformly  divided  scale,  divided  so  as  to  read 
amperes  directly.  Such  an  instrument  is  shewn  in  Fig.  226. 


Fig.  226. 


221.  Soft  iron  Ammeters.  In  another  class  of 
instruments  the  attraction  between  a  coil  carrying  a  current 
and  a  piece  of  soft  iron  is  used  to  measure  the  current.  A 
piece  of  soft  iron  is  arranged  so  that  it  can  be  sucked  up  into 
the  core  of  the  coil  when  the  current  passes  ;  the  distance 
the  iron  moves  is  indicated  by  the  motion  of  a  pointer 
attached  to  it.  The  pointer  moves  over  a  scale  and  the  current 
in  amperes  is  read  directly.  The  iron  is  prevented  by  its 
weight  from  being  sucked  completely  into  the  coil  and  the 
equilibrium  position  of  the  pointer  is  reached  when  the  couple 
due  to  the  attraction  between  the  coil  and  the  iron  balances 
that  due  to  the  weight  of  the  iron. 

If  the  susceptibility  of  iron  were  constant,  so  that  the 
induced  magnetisation  was  always  proportional  to  the  current, 
then  since  the  force  between  the  iron  and  the  coil  in  any 
given  position  depends  on  the  product  of  the  strength  of  the 
current  and  the  induced  magnetisation,  and  this  latter  is  by 
hypothesis  proportional  to  the  current,  the  force  will  depend 


364  ELECTRICITY  [CH.  XXII 

on  the  square  of  the  current ;  from  this  fact  the  instrument 
could  be  graduated,  but  since  the  susceptibility  is  not  constant, 
but  itself  depends  on  the  current,  such  an  instrument  has 
usually  to  be  graduated  by  direct  experiment,  employing 
calculations  based  on  the  magnetisation  curve  as  the  basis 
of  the  experiment.  Such  an  instrument  is  shewn  in  Fig.  227. 


Fig.  227. 


222.  Ammeters  and  Shunts.  When  an  ammeter  is 
used  for  the  measurement  of  a  large  current  only  a  small 
fraction  of  the  current  passes  through  the  coils  of  the  instru- 
ment. A  shunt  of  small  resistance  is  employed  to  connect 
together  the  poles  of  the  instrument,  and  the  main  portion 
of  the  current  flows  through  the  shunt.  A  definite  fraction 
of  the  current  depending  on  the  ratio  of  the  resistance  of 
the  shunt  to  that  of  the  coil,  passes  through  the  coil,  and  by 
measuring  this  the  whole  current  is  estimated. 

For  instance,  the  instrument  may  be  arranged  to  have  a 
resistance  of  1  ohm,  and  be  such  that  when  a  current  of  (H 
ampere  passes  through  the  coils  the  deflexion  is  100  scale 
divisions.  To  produce  a  current  of  O'l  ampere  a  potential 
difference  of  O'l  volt  is  required  in  this  case,  and  each 
division  of  the  scale  registers  the  passage  of  a  milliampere 
or  the  application  of  a  millivolt. 


221-222]          ELECTROMAGNETIC   INSTRUMENTS 


365 


Now  let  us  shunt  the  poles  A,  B  of  the  ammeter,  Fig.  228, 
by  1/99  of  an  ohm;  then  if  a  current  of  •]  ampere  enter 
at  A  one  hundredth  of  the  current  will  pass  through  the  coil 
C  and  99/100  through  the  shunt.  This  follows  from  Section  159, 


L/\/\/\/\/xJ 


Fig.  228. 

for  we  have  seen  that  if  S  be  the  resistance  of  the  shunt,  G 
that  of  the  galvanometer,  then  the  current  through  the 
galvanometer  coil  is  S/(S  +  G)  of  the  whole  current. 

Hence  in  the  case  supposed  the  sensitiveness  has  been 
reduced  one  hundredfold ;  the  deflexion  for  a  total  current 
of  O'l  ampere  would  only  be  one  division,  and  the  instrument 
would  read  up  to  100  x -1  amperes  or  10  amperes.  By  de- 
creasing the  resistance  of  the  shunt  we  increase  the  range. 
For  measuring  high  currents  we  should  probably  start  with  a 
less  sensitive  instrument,  one  in  which  for  example  an  E.M.F.  of 
a  volt  produced  a  deflexion  of  100  divisions,  then  if  the  shunt 
were  '001  ohms  in  resistance,  and  a  current  of  1000  amperes 
traversed  it,  the  E.M.F.  between  the  poles  would  be  1  volt 
and  the  deflexion  100.  The  total  current  would  be  rather 
greater  than  1000  amperes,  because  of  the  small  fraction  of 
the  whole  which  traverses  the  coil  of  the  ammeter,  but  by 
using  a  shunt  of  slightly  greater  value,  the  instrument  could 
be  adjusted  so  that  1000  amperes  corresponded  to  100  divisions; 


.366 


ELECTRICITY 


[CH.  XXII 


thus  its  range,  assuming  one  division  to  be  legible,  would  be 
from  10  to  1000  amperes. 

223.  Voltmeters.  The  same  instrument  can  be  used 
as  a  voltmeter.  For  let  us  take  the  first  case  when  the  coil 
resistance  is  1  ohm  and  the  deflection  for  O'l  volt  is  100. 
Place  99  ohms  in  series  with  the  coil  and  apply  10  volts  to 
the  ends  of  the  whole  resistance.  The  E.M.F.  between  the 
poles  will  still  be  0-1  volt,  the  current  O'l  ampere,  thus  the 
deflexion  is  still  100  divisions,  and  the  instrument  reads  from 
O'l  volt  up  to  10  volts.  Similarly  by  using  a  less  sensitive 
instrument  and  higher  resistances  we  can  measure  higher 
voltages.  Fig.  229  shews  such  an  instrument  with  its  shunts 


Fig.  229. 

and  resistances.  The  shunts,  one  of  which  is  shewn  to  the  left, 
are  strips  of  manganin  or  some  similar  resistance  material 
arranged  so  as  to  have  considerable  surface  and  therefore  to 
heat  but  little  with  the  passage  of  a  considerable  current ;  the 
volt  resistances  on  the  right  hand  of  the  figure  are  coils  wound 
as  in  an  ordinary  resistance  box. 

It  should  be  noticed  that  the  readings  of  such  an  instrument 
are  to  some  extent  affected  by  temperature.  They  depend  on 
the  ratio  of  two  resistances ;  one  of  these,  the  shunt  or  the 
volt  resistance,  is  usually  of  manganin  or  some  material  which 
does  not  change  much  in  resistance  with  temperature,  the 
other,  the  coil,  is  of  copper  and  has  a  considerable  temperature 
coefficient. 


222-224]          ELECTROMAGNETIC   INSTRUMENTS  367 

The  range  and  applicability  of  the  soft  iron  instruments 
can  be  extended  by  the  use  of  shunts  or  volt  resistances  in 
the  same  manner. 

224.  The  electrodynamometer.  In  this  instru- 
ment, which  may  take  various  forms,  one  of  which  is  shewn 
in  Fig.  230,  two  coils,  one  fixed,  the  other  moveable,  are  used. 


Fig.  230. 

The  centres  of  the  two  coils  coincide  and  their  axes  are  at 
right  angles  so  that  if  currents  circulate  in  the  two,  their 
lines  of  force  at  the  common  centre  are  perpendicular,  and 
the  moving  coil  tends  to  move  so  that  its  axis  and  its  field 
may  be  parallel  to  those  of  the  fixed  coil  respectively.  The 
moving  coil  is  suspended  in  such  a  way  as  to  resist  this  force, 
either  by  means  of  a  wire  having  torsion  or  by  a  bifilar 
suspension,  and  if  the  instrument  be  once  graduated,  then  by 
measuring  the  angle  through  which  the  coil  is  deflected  the 
current  can  be  calculated, 


368 


ELECTRICITY 


[CH.  XXII 


In  practice,  instead  of  allowing  the  coil  to  be  deflected  the 
suspension  head  is  turned  in  the  opposite  direction  to  that 
of  the  deflexion,  and  the  coil  is  thus  brought  back  to  its 
equilibrium  position.  The  angle  through  which  the  suspension 
head  is  turned  measures  the  couple  required  to  hold  the  coil  in 
its  equilibrium  position  and  enables  the  current  to  be  cal- 
culated. 

For  let  ily  i2  be  the  currents  in  the  two  coils  •  the  strength 
of  the  field  due  to  the  fixed  coil  is  proportional  to  ilt  the 
magnetic  moment  of  the  moving  coil  to  ?2,  and  the  couple  on 
a  magnet  of  moment  M  in  a  field  of  strength  ZTis  MH.  Hence 
the  couple  on  the  moving  coil  is  proportional  to  il  xi2.  Thus 
the  deflexions  are  proportional  to  ^  x  i2 . 

If  then  ij  be  a  known  constant  current  the  deflexions  will 
be  proportional  to  i2  and  the  current  can  be  measured.  More 
usually  however  the  same  current  i  circulates  in  the  two  coils, 
thus  t\  and  i,  are  equal,  and  the  deflexions  are  proportional 
to  i2.  Hence  i  the  current  is  measured  by  the  square  root 
of  the  deflexion. 


Fig.  231. 


225.     The   Ampere    balance.     This    instrument   in 
its  simple  form,    shewn    diagrammatically   in    Fig.    231,   has 


224-226]          ELECTROMAGNETIC    INSTRUMENTS  oOO 

a  coil  with  its  plane  horizontal  suspended  from  one  end  of  the 
beam  of  a  balance.  A  fixed  coil  with  its  plane  also  horizontal 
rests  below  this  and  the  weight  of  the  hanging  coil  is  counter- 
poised by  weights  in  the  pan.  Flexible  connexions  are 
arranged  so  that  a  current  can  be  passed  through  the  moveable 
coil ;  if  the  same  current  be  passed  through  the  fixed  coil  also 
there  is  an  attraction — or  it  may  be  a  repulsion — between  the 
two  which  is  proportional  to  the  product  of  the  moments  of 
the  two  equivalent  magnets.  As  each  of  these  is  proportional 
to  the  current  the  force  of  attraction  is  proportional  to  the 
square  of  the  current.  To  restore  equilibrium  weights  must 
be  placed  in  the  pan  of  the  balance,  and  the  current  in  any 
case  will  be  proportional  to  the  square  root  of  the  weight. 
Tims  two  currents  can  be  compared  by  comparing  the  square 
roots  of  the  weights  used  in  the  two  cases,  or  if  the  weight 
required  to  balance  some  given  current,  one  ampere  say,  be 
known  and  be  equal  to  TFJ,,  where  that  required  for  a  current 
i  is  found  to  be  IF,  then  i  is  given  by  the  equation 

W 

^  amperes. 
"  o 

The  standard  ampere  balance  of  the  Board  of  Trade  is 
constructed  in  this  manner ;  the  weight  required  to  produce 
equilibrium  when  a  current  of  one  ampere  is  circulating  in 
the  coils  was  determined  with  great  care  and  so  long  as  the 
construction  of  the  balance  does  not  change  in  any  way  this 
weight  remains  fixed. 

The  ampere  is  now  defined  for  legal  purposes  as  the  current 
which  must  be  passed  through  the  coils  of  the  instrument  to 
balance  this  weight. 

226.  Lord  Kelvin's  Ampere  balance.  The  most 
usual  form  of  current  balance,  however,  is  that  of  Lord  Kelvin. 

One  of  these  is  shewn  in  Figs.  232  (a)  and  (6)  of  which 
Fig.  232  (a)  gives  the  connexions  and  indicates  the  path  of  the 
current  while  232  (6)  is  a  drawing  of  the  complete  instrument. 
Two  moveable  coils  are  attached,  one  to  each  end  of  a  balance 
beam ;  both  beneath  and  above  these  are  fixed  coils,  and  the 
direction  of  the  current  is  such  that  the  electromagnetic  force 

G.  E.  24 


370 


ELECTRICITY 


[CH.  XXII 


on  one  of  the  moveable  coils  is  upwards  while  on  the  other  it 
is  downwards. 


Fig.  232  a. 

A  moveable  weight  can  slide  on  a  graduated  horizontal 
arm  attached  to  the  beam,  and  a  pointer  at  the  end  of  this 
arm  shews  when  the  beam  is  in  its  horizontal  position. 


On  passing  the  current  through  the  coils  the  beam  is 
deflected,  and  the  weight  must  be  shifted  to  bring  the  pointer 
back  to  its  sighted  position. 

The  couple  deflecting  the  beam  is  proportional  to  the 
square  of  the  current,  that  restoring  it  is  proportional  to  the 
distance  the  weight  has  been  shifted  from  the  fulcrum. 


226-228]          ELECTROMAGNETIC   INSTRUMENTS  371 

Hence  for  a  given  weight,  the  current  is  proportional  to 
the  square  root  of  the  distance  between  the  weight  and  the 
fulcrum,  and  by  graduating  the  arm  along  which  the  weight 
slides  according  to  a  square  root  scale,  the  currents  can  be 
read  directly  from  the  position  of  the  weight. 

The  current  is  introduced  into  the  instrument  by  flexible 
leads,  which  do  not  interfere  with  the  action.  In  Lord 
Kelvin's  instruments,  these  form  the  ligaments  by  which  the 
beam  is  suspended.  The  value  of  the  weight  is  of  course 
determined  by  an  experiment  with  a  known  current ;  by 
employing  a  series  of  weights  the  range  of  the  instrument  is 
extended. 


227.  Measurement  of  Alternating  Current.     It 
should   be   noticed   that  the  indications  of    these  last  three 
instruments  depend  on  the  square  of  the  current.     Thus  by 
reversing  the  current,  so  long  as  its  magnitude  is  unaltered, 
the  indication  of  the  instrument  is  unaffected.     In  many  ap- 
plications of  electricity  alternating  currents  (see  Section  246) 
in  which  the  direction  of  the  current  is  being  continually 
reversed  are  employed.     These  instruments  may  with  proper 
precautions  be  used  to  measure  such  currents. 

228.  Influence  of  external  fields.     The  indications 
of   any    of   the   above  instruments  will   be  affected  to  some 
extent  by  the  strength  of  the  external  magnetic  field  in  which 
they  happen  to  be  placed  ;  in  the  case  of  a  moving  coil  galva- 
nometer, for  example,  the  deflexion  depends  on  the  strength 
of  the  field  in  which  the  coil  hangs,  and  so  with  the  others ; 
but  since  the  field  of  the  permanent  magnets  is  very  intense 
compared  with  the  earth's  field,  for  example — (in  a  good  in- 
strument the  field  may  be  800  to  1000  units,  while  the  earth's 
field  is  *18) — the  effect  of  the  earth's  field  is  very  small,  and 
that  of  small  changes  of  that  field   is  smaller  still.     While 
small  changes  in  the  direction  of  the  earth's  horizontal  force 
produce  a  considerable  effect  on  the  zero  of  an  ordinary  astatic 
galvanometer,  they  are  practically  inappreciable  with  a  sus- 
pended coil  instrument.     Indeed  such  an  instrument  can  be 
employed  in  the  neighbourhood  of  electromagnetic  machines, 

24—2 


372  ELECTRICITY  [CH.  XXII 

such  as  dynamos  or  motors  (see  Section  240)  without  its 
indications  being  very  seriously  affected,  in  positions  in  which 
an  ordinary  moving  magnet  galvanometer  would  be  useless. 
The  reason  is  clear ;  in  the  suspended  coil  instrument  the 
deflexion  is  increased  by  increasing  the  field  strength  of  the 
instrument.  This  therefore  is  made  so  strong  that  external 
changes  are  masked.  In  a  moving  magnet  instrument  the 
sensitiveness  is  increased  by  weakening  the  tield  of  the 
instrument,  small  changes,  then,  in  the  weak  field  due  to 
external  influences  produce  large  effects. 


CHAPTER  XXIII. 

ELECTROMAGNETIC   INDUCTION. 

229.     Faraday's  Experiments  on  Induction.     We 

have  already  seen,  Section  199,  that  when  a  circuit  carrying 
a  current  is  placed  in  a  magnetic  field  a  force  acts  on  it  and 
tends  to  move  it  into  such  a  position  that  a  maximum  number 
of  lines  of  induction  due  to  the  field  may  pass  through  the 
circuit  from  its  south  face  towards  its  north  face. 

We  proceed  now  to  deal  with  some  phenomena,  first 
observed  by  Faraday,  which  may  be  looked  upon  as  the 
converse  of  the  above. 

Faraday  shewed  that  whenever  a  circuit  in  a  magnetic 
field  was  moved  relatively  to  the  field  so  as  to  vary  the 
number  of  lines  of  magnetic  induction  which  traverse  it,  then 
an  electromotive  force  is  set  up  round  the  circuit  causing  a 
current  to  flow  which  lasts  so  long  as  the  number  of  lines  of 
induction  linked  with  the  circuit  is  varied.  The  relative 
motion  of  the  field  and  the  circuit  may  be  produced  by  moving 
either  the  circuit  or  the  magnetic  system  to  which  the  field  is 
due. 

Moreover  it  should  be  noted  that  the  currents  set  up  are 
always  such  as  to  oppose  by  their  electromagnetic  action  on 
the  field  the  motion  to  which  they  are  due. 

These  currents  are  said  to  be  induced. 

EXPERIMENT  54.  To  investigate  the  production  of  an  in- 
duced current  by  the  motion  of  a  magnet. 

The  ends  of  a  coil  of  wire,  Fig.  233,  are  connected  to  a 
galvanometer  which  is  placed  at  some  distance  from  the  coil. 

A  galvanometer  may  if  necessary  be  constructed  for  the  purpose  by 
utilizing  a  similar  coil.  This  is  placed  in  the  magnetic  meridian  and  a 


374 


ELECTRICITY 


[CH.  XXIII 


compass  needle  is  pivoted  at  its  centre ;  when  undisturbed  the  compass- 
needle  rests  with  its  axis  in  the  plane  of  the  coil,  on  passing  a  current 
through  the  coil  the  needle  is  deflected. 


Fig.  233. 

Take  a  long  bar  magnet  and  move  its  north  pole  up  to  the 
coil  of  wire,  passing  it  through  the  coil,  the  galvanometer 
needle  is  deflected  shewing  a  current  through  the  circuit,  as 
soon  however  as  the  motion  of  the  magnet  ceases  the  needle 
comes  back  to  its  zero  position,  thus  the  current  only  lasts 
while  the  motion  continues. 

Withdraw  the  magnet  from  the  coil.  The  galvanometer 
needle  is  again  deflected,  but  in  the  opposite  direction. 

Repeat  the  experiment,  but  now  bring  up  the  south  pole 
of  the  magnet  instead  of  the  north,  the  galvanometer  needle  is 
deflected  as  in  the  last  case ;  the  approach  of  the  south  pole 
therefore  causes  a  current  in  the  same  direction  as  the  with- 
drawal of  the  north  pole  and  vice  versd. 

The  direction  of  the  current  can  be  inferred  from  the 
direction  of  motion  of  the  magnet.  To  make  things  clear 
suppose  the  coil  placed  with  its  plane  vertical  and  north  and 
south  and  connected  with  the  galvanometer  coil  in  such  a  way 
that  the  current  may  circulate  in  the  same  direction  round 
the  two,  and  let  the  north  pole  of  the  magnet  approach  the 
coil  from  the  east  side. 


229] 


ELECTROMAGNETIC   INDUCTION 


375 


Then  it  will  be  found  that  the  north  pole  of  the 
galvanometer  needle  is  deflected  to  the  east.  From  this  we 
infer  that  the  current  in  both  coils  passes  from  south  to  north 
below  the  magnet,  Fig.  234.  Thus  if  the  plane  of  the  paper 
represent  the  face  of  the  coil  of  wire  which  the  magnet 

TOP 


BOTTOM 
Fig.  234. 

approaches,  so  that  the  upper  side  of  the  paper  is  east,  the 
right  hand  being  north,  the  current  is  as  indicated  in  the 
figure.  Thus  the  upper  surface  of  the  paper  becomes  the 
north  face  of  the  magnet  which  is  equivalent  to  the  circuit, 
and  the  motion  of  the  magnet  inducing  the  current  is 
opposed  by  the  repulsion  between  the  circuit  and  itself. 

On  withdrawing  the  north  pole  it  will  be  found  that  the 
direction  of  the  current  is  reversed,  the  upper  face  of  the 
circuit  is  a  south  face  and  the  north  pole  is  attracted  by  this, 
the  electromagnetic  forces  again  oppose  the  motion. 

It  is  clear  from  the  principle  of  the  conservation  of  energy 
that  this  must  be  so;  for  the  current  possesses  energy  and 
this  energy  it  obtains  from  the  work  done  in  moving  the 
magnet  towards  the  circuit  in  opposition  to  the  electromagnetic 
repulsion,  or  withdrawing  it  from  the  circuit  in  opposition 
to  the  electromagnetic  attraction. 

Corresponding  effects  are  produced  if  the  magnet  be  kept 
fixed  and  the  coil  moved  up  to  it. 

The  above  effects  have  been  produced  by  the  motion  of  a 
magnet.  For  the  magnet  we  may  substitute  a  coil  of  wire 
carrying  a  current. 


376 


ELECTRICITY 


[CH.  XXIII 


EXPERIMENT  55.  To  investigate  tlie  production  of  an  in- 
duced current  by  the  change  in  the  strength  of  a  neighbouring 
current. 

The  same  apparatus  is  required  as  in  the  preceding  experi- 
ment, except  that  for  the  magnet  we  substitute  another  coil  of 
wire  I.  connected  to  a  battery  and  to  a  key,  Fig.  235.  This 


Y 


Fig.  235. 

second  coil  is  placed  near  to  the  first,  the  planes  of  the  two 
being  parallel.  As  previously,  the  coils  I.  and  II.  should  be 
placed  at  some  distance  from  the  galvanometer  G  so  as  to 
avoid  any  direct  magnetic  action  on  its  needle.  The  coil 
connected  to  the  battery  is  spoken  of  as  the  primary  coil, 
that  connected  to  the  galvanometer  as  the  secondary  coil. 

Resistance  boxes  X  and  Y  are  conveniently  included  in 
the  circuits  in  order  to  vary  the  currents  if  desired.  The 
primary  circuit  also  conveniently  includes  an  ammeter  A. 

Make  contact  in  the  primary  circuit  and  allow  a  current 
to  pass ;  the  galvanometer  needle  is  temporarily  deflected,  but 
if  the  key  be  held  down  it  returns  at  once  to  its  zero  position. 
Break  contact  at  the  key ;  the  needle  is  again  temporarily 
deflected,  but  in  the  opposite  direction  to  its  previous  motion. 

The  first  contact  has  produced  a  magnetic  field  in  the 
neighbourhood  of  the  primary  coil.  Lines  of  magnetic  induc- 
tion well  out  from  the  coil,  some  of  these  thread  the  secondary, 


229-230]  ELECTROMAGNETIC    INDUCTION  877 

thus  the  number  of  lines  of  induction  linked  with  the  secondary 
is  varied  and  an  induced  current  is  the  consequence. 

If  the  distance  between  the  two  coils  be  increased  the 
effect  is  diminished,  if  on  the  other  hand  a  bundle  of  iron 
rods  is  inserted  in  the  secondary,  the  effect  is  to  increase 
greatly  the  induction  through  it,  and  the  induced  currents 
are  larger.  Non-magnetic  materials  placed  in  the  coils  do 
not  change  the  result. 

In  some  cases  we  can  calculate  the  total  change  of  induc- 
tion through  the  secondary ;  if  the  galvanometer  be  a  suitable 
one  arranged  for  ballistic  work,  we  have  seen  already,  when 
treating  of  condensers,  that  the  first  throw  of  the  needle  is  pro- 
portional to  the  total  quantity  of  electricity  which  circulates 
round  the  coil,  and  it  is  not  difficult  to  arrange  experiments 
to  prove  that  this  total  quantity  is  proportional  to  the  total 
change  of  induction  and  inversely  proportional  to  the  total 
resistance  of  the  circuit1. 

Thus  the  total  R.M.F.  round  the  secondary  is  proportional 
to  the  total  change  in  the  number  of  lines  of  induction  linked 
with  it. 

230.     Coefficient   of  Mutual   Induction.     Let  us 

suppose  a  unit  current  is  circulating  in  the  primary,  then  a 
certain  number  of  lines  of  induction  issue  from  it  and  of 
these  some  are  linked  with  the  secondary.  This  number  will 
depend  on  the  dimensions  and  relative  position  of  the  two 
circuits  ;  it  is  called  the  Coefficient  of  Mutual  Induction 
of  the  two  circuits.  Let  us  denote  it  by  M ;  then  if  we  double 
the  current  in  the  primary  we  double  the  induction — supposing 
the  field  free  from  iron — everywhere.  Thus  the  number  of 
lines  of  induction  through  the  secondary  is  doubled,  and  if  a 
current  i  circulates  in  the  primary,  the  total  induction  through 
the  secondary  is  Mi. 

In  estimating  the  number  of  lines  of  induction  linked  with 
the  circuit,  the  number  of  turns  in  the  circuit  must  be 
considered.  If  MQ  lines  pass  through  each  turn  and  the 
number  of  turns  is  m,  then  the  total  number  of  linkages 

1  Glazebrook  and  Sha\v,  Practical  Physics. 


378  ELECTRICITY  [CH.  XXIII 

is  mM0.  If  unit  current  be  started  in  the  primary  there 
will  be  an  B.M.F.  J/0  in  each  of  the  m  turns  of  the  secondary, 
thus  the  total  E.M.P.  in  the  secondary  is  m  x  J/0. 

It  can  also  be  shewn  that  if  a  unit  current  circulates  in 
the  secondary,  the  number  of  lines  of  induction  which  thread 
the  primary  is  also  M. 

Thus  we  have  the  following 

DEFINITION.  The  number  of  lines  of  induction  due  to  unit 
current  in  one  circuit  which  thread  a  second  is  known  as  the 
Coefficient  of  Mutual  Induction  between  the  two. 

It  follows  from  the  above  that  if  the  coils  be  moved  so  that  their 
coefficient  of  mutual  induction  is  changed  from  M  to  M',  and  if  simul- 
taneously the  current  in  the  primary  changes  from  i  to  i',  then  the  total 
electromotive  force  in  the  secondary  is  Mi  -  M'i'  and  the  total  flux  of 
electricity  round  it  is  (Mi-M'i')IE,  where  R  is  the  resistance  of  the 
secondary. 

231.  Lenz's   Law.     The  law  to  which  reference  has 
already  been  made  which  states  that  in  all  cases  of  electro- 
magnetic induction  the  induced  currents  have  such  a  direction 
that  their  reactions  tend  to  stop  the  motion  to  which  they  are 
due  is  known  as  Lenz's  Law. 

232.  Self-induction.      When    a    current    is   started 
round  a  coil  of  wire  the  number  of  lines  of  induction  through 
that  coil  is  varied.     In  consequence  an  induced  electromotive 
force  tending  to  oppose  the  current  is  produced,  the  current 
rises  to  its  steady  value  less  rapidly  than  it  otherwise  would ; 
some  of  the  energy  supplied  by  the  battery  goes  to  establish  a 
magnetic  field  in  the  neighbourhood. 

When  the  current  is  broken,  the  number  of-  lines  of 
induction  through  the  circuit  is  reduced,  an  electromotive 
force  is  set  up  acting  in  the  same  direction  as  the  original 
electromotive  force  tending  to  maintain  the  original  current. 

This  is  called  the  electromotive  force  of  self-induction. 

DEFINITION.  The  number  of  lines  of  induction  due  to  unit 
current  in  a  circuit  which  are  linked  with  the  circuit  is  called 
the  Coefficient  of  Self-induction  of  the  circuit. 


230-234]  ELECTROMAGNETIC    INDUCTION  379 

Hence  if  L  be  the  coefficient  of  self-induction  of  a  circuit 
carrying  a  current  i  and  having  a  resistance  R,  then  the  total 
E.M.F.  of  self-induction  round  the  circuit  is  Li  and  the  total 
flux  of  electricity  it  causes  is  LifR. 

233.  Unit   of  Inductance.     The   practical   unit   in 
terms  of   which  coefficients  of   self  or  mutual  induction  are 
measured  is  called  the  Henry. 

DEFINITION.  The  coefficient  of  mutual  induction  between 
two  circuits  is  one  Henry  when,  if  a  current  of  1  ampere  is 
passed  round  the  primary  circuit,  the  flux  of  electricity  round 
the  secondary  of  resistance  R  ohms  is  that  carried  by  a  current 
of  \IR  amperes  flowing  for  one  second. 

234.  Observations  on  Induction. 

EXPERIMENT  56.  To  shew  the  production  of  the  current 
of  self-induction. 

(1)  A  battery  is  connected  through  a  key  with  a  coil  of 
many  turns  having  a  large  coefficient  of  self-induction — a  coil 
with  an  iron  core  is  usually  employed.  The  ends  of  the  coil, 
AB,  Fig.  236,  are  also  connected  through  a  rough  galvanometer. 


B 

Fig.  236. 

When  the  key  is  closed  a  current  passes  through  the  coil  and 
the  galvanometer  traversing  both  in  the  direction  from  A  to  R 
suppose ;  let  it  deflect  the  north  end  of  the  galvanometer  needle 
to  the  right  and  place  a  stop  in  contact  with  the  needle  so 
that  this  deflexion  cannot  take  place. 

Break  the  circuit  at  the  key.     An  induced  electromotive 
force  acts  through  the  coil  from  A  to  h  in  the  same  direction 


380  ELECTKICITY  [CH.  XXIII 

as  the  original  E.M.F.  ;  the  induced  current  now  passes  through 
the  galvanometer  from  B  to  A  and  the  north  end  of  the  needle 
is  deflected  to  the  left. 

(2)  The  same  effect  may  be  shewn  better  with  the  aid  of 
an  incandescent  lamp  in  place  of  this  galvanometer.  The 
battery  can  be  adjusted  so  that  the  steady  current  is  insufficient 
to  glow  the  lamp ;  on  breaking  circuit  it  glows  instantaneously, 
the  much  more  powerful  electromotive  force  of  self-induction 
drives  sufficient  current  through  it  to  render  it  incandescent; 
if  the  current  be  again  made  it  glows  again  for  an  instant; 
the  E.M.F.  of  self-induction  at  make  checks  the  current  in  the 
coil  and  causes  a  large  flow  through  the  lamp;  by  arranging 
an  interrupter  in  the  circuit  and  working  it  continuously  the 
lamp  may  be  kept  in  a  steady  state  of  glow. 

EXPERIMENT  57.  To  shew  the  production  of  an  induced 
current  due  to  the  motion  of  a  coil  in  the  earths  field. 

Connect  a  coil  resting  on  the  table,  Fig.  237,  to  a  somewhat 
sensitive  galvanometer.  Lift  it  up  quickly  and  reverse  its 


Fig.  237. 

position,  laying  it  down  with  the  bottom  face  uppermost;  the 
galvanometer  shews  an  instantaneous  current. 

The  lines  of  magnetic  induction  due  to  the  earth  traverse 
the  coil  in  a  direction  opposite  to  their  previous  one.;  this 
change  has  caused  an  induced  current  in  the  coil. 

(3)  Arrange  the  coil  as  shewn  in  Fig.  238,  s.o  that  it  can 
rotate  about  an  axis  in  its  own  plane.  Turn  the  coil  about 
this  axis  through  180°.  The  galvanometer  needle  is  deflected, 
but  returns  to  zero  again;  an  induced  current  has  passed; 


234] 


EL ECTROM AGN ETIC    IN DUCTION 


381 


continue  the  rotation  through  a  further  180° ;  the  needle  is 
again  deflected  but  in  the  opposite  direction  ;  a  second  induced 
current  opposite  in  direction  to  the  first  has  traversed  the 
circuit;  thus  by  continuing  the  rotation  we  get  an  alternating 
current  in  the  circuit. 


Fig.  238. 

If  we  call  X  and  Y  the  two  faces  of  the  coil  and  suppose 
that  in  the  original  position  a  maximum  number  of  lines 
of  magnetic  induction  due  to  the  earth  enter  the  coil  at  the 
face  X  and  leave  it  at  the  face  Y,  as  the  coil  rotates  the 
number  of  lines  entering  at  X  decreases  till  at  last  it  vanishes. 
As  the  rotation  continues  lines  of  induction  begin  to  enter  at 
Y  increasing  up  to  a  maximum  when  a  rotation  through  180° 
has  been  completed.  Throughout  this  part  of  the  rotation 
the  E.M.F.  has  the  same  direction,  the  lines  of  force  entering  X 
are  decreasing,  those  entering  at  Y  are  increasing ;  and  these 
two  effects  contribute  to  the  same  result ;  as  the  rotation 
continues  through  a  further  180°  the  reverse  is  the  case,  the 
lines  entering  Y  decrease,  those  entering  ^Y  increase  ;  thus  the 
induced  current  changes  its  direction,  to  reverse  again  when 
the  first  position  is  reached1. 

We    can   obtain   by  calculation   a    relation    between   the 

1  When  the  effects  of  self-induction  are  considered  this  statement 
needs  a  slight  modification  which  is  however  not  important  for  our 
present  purpose. 


382  ELECTRICITY  [CH.  XXIII 

induced  current  and  the  strength  of  the  field  in  which  the 
coil  is  rotated  and  hence  by  measuring  the  current  can 
calculate  the  strength  of  the  field.  This  is  made  use  of  in  the 
earth  inductor. 

By  attaching  a  split  ring  commutator  to  the  axis  of  the 
coil  we  can  divert  all  the  currents  into  the  same  direction 
in  the  external  circuit. 

The  arrangement  has  been  already  described  in  Section  157 
and  as  adapted  to  the  present  purpose  is  illustrated  in  Fig.  245  «. 
The  two  halves  of  the  split  ring  are  mounted  on  the  axis 
about  which  the  coil  rotates,  being  insulated  from  each  other, 
and  the  ends  of  the  coil  are  connected  to  these. 

Two  springs  to  which  the  ends  of  the  external  circuit 
are  connected  press  against  the  split  ring.  In  one  position 
a  current  circulating  in  the  coil  in  a  given  direction 
passes  round  the  external  circuit  from  Q  to  P.  As  the  coil 
rotates  the  connexions  at  the  commutator  are  reversed ;  the 
commutator  is  so  fixed  to  the  coil  that  at  the  same  moment 
as  this  occurs  the  direction  of  the  current  is  reversed  in 
the  coil ;  hence  its  direction  in  the  external  circuit  will  still 
be  from  Q  to  P.  Thus  by  the  aid  of  the  commutator  the 
currents  in  the  external  circuit  are  all  diverted  into  the  same 
direction. 

Such  an  arrangement  of  a  coil  rotating  in  a  magnetic  field 
and  thus  producing  a  current  constitutes  the  simplest  form  of 
an  electromagnetic  machine.  See  Section  240. 

235.  Arago's  Disc.  If  a  conductor  such  as  a  copper 
disc  be  moved  near  a  magnet,  induction  currents  are  set  up  in 
the  conductor ;  these  since  they  circulate  in  closed  curves  in  the 
substance  of  the  conductor  are  known  as  eddy  currents.  The 
electromagnetic  forces  called  into  play  tend  to  stop  the  disc ; 
they  tend  therefore  to  move  the  magnet  in  the  same  direction 
as  the  disc. 

This  is  illustrated  in  Arago's  apparatus,  Fig.  239.  A 
copper  disc  is  made  to  rotate  about  a  vertical  axis  and  a  bar 
magnet  is  pivoted  above  the  centre  of  the  disc ;  the  magnet 
is  separated  from  the  disc  by  a  sheet  of  glass  which  protects 
it  from  air  currents. 


234-236]  ELECTROMAGNETIC    INDUCTION 


383 


On  setting  the  disc  in  motion  the  induced  currents 
produced  act  on  the  magnet  and  drag  it  round  after  the  disc, 
its  speed  gradually  increases  until  it  is  moving  at  the  same 
rate  as  the  disc ;  when  this  is  the  case  the  disc  no  longer  cuts 


TT 


Fig.  239. 

lines  of  induction  due  to  the  magnet  and  the  induced  currents 
cease.  If  the  disc  be  now  stopped,  with  the  magnet  still  moving, 
induced  currents  are  again  started  and  these  act  as  a  drag  on 
the  magnet,  gradually  bringing  it  to  rest. 

In  a  somewhat  different  form  of  the  experiment  the  disc 
is  mounted  so  that  it  can  rotate  between  the  poles  of  an 
electromagnet. 

Before  the  electromagnet  is  excited  the  disc  turns  easily  ; 
when  the  magnet  is  excited  induced  currents  are  set  up  in  the 
disc  and  the  forces  called  into  play  tend  to  stop  the  motion ; 
much  force  is  needed  to  keep  the  disc  rotating  while  the  disc 
becomes  strongly  heated  by  the  eddy  currents  which  circulate 
in  it. 

236.  Electromagnetic  method  of  measuring 
permeability.  As  we  have  already  seen  the  total  quantity 
of  electricity  which  circulates  round  a  secondary  circuit  is 
equal  to  the  ratio  of  the  change  in  the  number  of  lines  of 


384  ELECTRICITY  [CH.  XXIII 

induction  linked  with  the  circuit  to  the  resistance  of  the 
circuit.  The  quantity  of  electricity  can  be  measured  as  in 
Section  174  by  means  of  a  ballistic  galvanometer  included  in 
the  secondary  circuit  ;  hence  by  multiplying  this  by  the  resist- 
ance we  can  find  the  total  change  of  induction  through  the 
circuit.  On  this  are  founded  several  methods  for  measuring 
the  permeability  of  iron  or  other  magnetic  materials. 

Thus,  if  the  iron  take  the  form  of  a  rod,  we  may  wind  a 
long    coil    round    it   as    in    Fig.    240.     This    constitutes    the 


Fig.  240. 

primary  coil,  and  the  iron  can  be  subjected  to  a  known  mag- 
netising force,  by  sending  a  current  of  measured  strength 
through  the  coil  ;  the  current  is  measured  by  including  an 
ammeter  in  the  circuit. 

A  few  turns  of  insulated  wire  wound  round  the  coil  and 
bar  near  its  centre  constitute  the  secondary  coil  ;  these  are 
connected  to  a  suitable  ballistic  galvanometer;  by  means  of 
some  subsidiary  experiments  the  relation  between  .the  de- 
flexion of  this  galvanometer  and  the  quantity  of  electricity 
which  circulates  in  the  circuit  is  determined  ;  thus,  when  this 
is  done,  by  observing  the  throw  of  the  galvanometer  needle 
the  total  flux  of  electricity  can  be  found. 

If  now  the  primary  circuit  be  closed,  the  iron  is  subject  to 
a  definite  magnetising  force,  and  the  galvanometer  needle 
disturbed  ;  the  total  induction  in  the  iron  is  determined  by 
calculating  from  the  induction  throw  the  total  flux  of  elec- 
tricity in  the  secondary  and  multiplying  this  by  the  resistance. 
Knowing  then  the  induction  and  the  magnetising  force,  we 
can  find  the  permeability.  The  experiment  in  this  form  is 
open  to  some  objections  for  poles  are  formed  near  the  end  of 
the  rod  and  the  magnetising  force  in  the  rod  is  modified  by 
their  presence.  Moreover  lines  of  force  pass  back  through 
the  air  round  the  rod,  and  since  the  secondary  coil  cannot 
be  in  contact  with  the  iron  some  of  these  lines  of  force 


236] 


ELECTKOMAGNETIC    INDUCTION 


385 


pass  through  it  and  modify  the  result.  The  total  induction 
linked  with  the  coil  is  the  induction  in  the  iron  together  with 
that  in  the  air  surrounding  the  iron. 

These  difficulties  are  overcome  by  forming  the  iron  into 
a  closed  ring,  and  winding  the  primary  coil  uniformly  round 
it.  In  this  case  the  lines  of  induction  lie  within  the  iron 
while  the  magnetic  force  due  to  a  current  i  is  kmrijl  where 
I  is  the  length  of  the  axis  of  the  ring  and  n  the  number  of 
turns  on  it. 

The  secondary  consists  as  before  of  a  few  turns  of  wire 
connected  to  a  ballistic  galvanometer  and  the  induction  is 
measured  by  the  throw  of  this  instrument. 

A  rheostat  is  included  in  the  primary  circuit  and  by 
means  of  this  the  strength  of  the  current  and  therefore  of  the 
magnetising  force  can  be  adjusted.  The  arrangement  of 


Fig.  241. 

apparatus  is  shewn  iri  Fig.  "241  in  which  (7  is  the  ring  with 
the  two  coils  wound  on  it,  B  the  battery,  A  an  ampere  meter 
for  the  primary  current,  R  the  rheostat  and  K  a  reversing 
key.  The  induced  current  is  measured  by  the  ballistic  gal- 
vanometer G. 

By  means  of  this  arrangement  magnetisation  and  hys- 
teresis curves  can  be  obtained  as  in  Section  216. 

The  iron  is  first  demagnetised  by  rapidly  reversing  and 
gradually  reducing  the  current.  A  suitable  resistance  is 

G.  E.  25 


386 


ELECTRICITY 


[CH.  XXIII 


taken  out  of  the  primary  arid  the  current  made,  the  corre- 
sponding throw  of  the  galvanometer  gives  the  induction ;  the 
iron  is  again  demagnetised,  the  primary  resistance  decreased 
and  the  current  again  made,  thus  a  second  value  of  the 


Fig.  242. 

induction  is  found  and  the  curve  of  induction  can  be  drawn 
as  shewn  in  Fig.  242  for  a  magnetising  force  increasing 
from  zero. 

To  form  the  descending  part  of  the  curve  we  start  with 
the  iron  fully  magnetised  and  suddenly  reduce  the  current 
by  a  known  amount.  The  induction  throw  observed  gives 
us  the  change  in  induction,  and  subtracting  this  from  the 
induction  due  to  the  full  current,  we  can  find  the  point  on 
the  descending  branch  which  corresponds  to  the  value  to 
which  the  current  was  reduced.  The  curve  thus  obtained 
agrees  with  that  obtained  by  the  magnetometric  method. 


EXAMPLES  387 


EXAMPLES   ON  ELECTROMAGNETISM  AND 
INDUCTION  OF  CURRENTS. 

1.  A  magnet  is  suspended  at  the  centre  of  a  coil  of  wire,  the  coil  being 
placed  in  an  east  and  west  vertical  plane.     When  there  is  no  current 
passing,  the   magnet   oscillates   15  times  in   10  seconds,    and   when   a 
current  is  passing  the  swings  number  25  in  10  seconds.     Find  the  field 
produced  by  the  current. 

(H=-1S  C.G.S.  unit.) 

2.  A  magnet  is  placed  in  a  coil  with  its  axis  parallel  to  the  axis 
of  the  coil.     It  is  then  quickly  withdrawn.     Will  the  direction  of  the 
current  in  the  coil  depend  upon  the  direction  in  which  the  magnet  is 
drawn  out  of  it? 

3.  The  ends  of  a  circular  coil  of  wire  are  connected  with  a  galvano- 
meter and  the  coil  can  be  made  to  rotate  about  one  of  its  diameters.    On 
turning  the  coil  half  round,   the  galvanometer  needle   is  momentarily 
deflected,  but  on  causing  it  to  rotate  in  one  direction  continuously  and 
rapidly  no  effect  is  produced.     Explain  these  two  results. 

4.  A  plane  rectangular  iron  frame  is  placed  vertically  so  that  it  faces 
due  magnetic  north.     It  is  made  to  fall  northward  into  a  horizontal 
position.     Describe  what  will  be  the  result. 

5.  A  circular  coil  of  wire  spins  about  a  diameter  which  is  placed 
vertically  (1)  at  the  magnetic  equator  and  (2)  at  the  magnetic  pole. 
Explain  the  nature  of  the  electric  currents,  if  any,  which  will  flow  round 
the  coil  in  each  case. 

6.  Two  coils  A  and  B  are  placed  parallel  to  each  other,  the  former 
being   connected   to    a   battery   and   key,    and   the   latter   to  a  ballistic 
galvanometer.     Shew   that,    if  the   galvanometer   has   a   very  high   re- 
sistance, the  throw  of  the  needle  on  starting  a  current  in  A  is  approxi- 
mately   proportional    to    the    number    of    turns    in   B,    whilst    if    the 
galvanometer  resistance  is  very  small  the  throw  is  nearly  independent 
of  the  number  of  turns. 

7.  A  very  long  magnet,  lying  along  the  axis  of   a  coil  of   wire,  is 
moved  in  the  direction  of  its  length  with  a  velocity  of  10  cm.  per  second. 
The  pole  strength  of  the  magnet  is    5  C.G.S.  units.     Find  in  volts  the 
greatest  value  of  the  E.M.F.  induced  in  the  coil,  which  has  a  mean  radius 
of  14  cm.  and  possesses  800  turns. 

8.  Twenty  turns  of  wire  are  wound  round  a  square  frame,  each  side  of 
which  is  30  cm.,  and  the  whole  is  rotated  about  a  diameter  in  a  magnetic 
field  of  500  C.G.S.  units  strength.     Find  the  number  of  revolutions  per 
second  in  order  that  the  average  E.M.F.  in  the  wire  may  be  1  volt. 


25—2 


CHAPTER  XXIV. 

APPLICATIONS   OF  ELECTROMAGNETIC   INDUCTION. 

237.  Principle  of  Transformers.  The  phenomena 
of  induction  are  made  use  of  in  induction  coils  and  trans- 
formers to  enable  us  to  produce  from  a  given  current  one 
of  a  different  electromotive  foroe. 

Consider  a  coil  of  a  moderate  number  of  turns  of  thick 
insulated  wire  through  which  a  current  can  be  passed  arranged 
as  in  Fig.  240  above.  This  is  called  the  primary  coil. 
Let  there  be  wound  over  this  m  times  as  many  coils  of  thinner 
wire— the  secondary  coil.  Tf  a  current  circulate  in  the 
primary,  a  certain  number  of  lines  of  induction  pass  through 
it ;  these  are  also  linked  with  the  secondary,  but  since  the 
number  of  turns  on  the  secondary  is  m  times  as  great  as 
that  on  the  primary  the  number  of  linkages  is  m  times  as 
great ;  if  the  primary  current  be  now  broken,  induced  electro- 
motive forces  are  set  up  in  both  coils  but  the  E.M.F.  of  induc- 
tion in  the  secondary  will  be  m  times  as  great  as  in  the 
primary;  thus  by  breaking  the  current  in  the  primary  we 
get  an  induced  current  of  much  greater  electromotive  force 
in  the  secondary  ;  if  now  we  make  the  primary  circuit,  an 
induced  current  of  great  electromotive  force  is  set  up  in  the 
opposite  direction  in  the  secondary  ;  and  by  alternately  making 
and  breaking  the  primary  alternating  induced  currents  of 
high  electromotive  force  are  produced  in  the  secondary. 

The  effect  is  increased  by  introducing  soft  iron  into  the 
core  of  the  coils  for  in  this  way  the  total  induction  is  greatly 
increased. 


237-238]  ELECTROMAGNETIC   INDUCTION 


389 


Again  the  total  flux  of  electricity  in  the  secondary  depends 
on  the  total  change  of  induction  and  is  therefore  the  same 
whether  the  break  in  the  primary  he  sudden  or  prolonged. 
But  the  time  during  which  this  flux  lasts  is  reduced  by 
making  the  break  a  sudden  one;  thus  the  strength  of  the 
current  in  the  secondary  at  each  instant  while  it  lasts  is 
increased  by  increasing  the  suddenness  of  the  break  or  make. 

238.  The  Induction  Coil.  The  induction  coil  usually 
takes  the  form  shewn  in  Fig.  243.  The  iron  core  consists  of 
a  bundle  of  soft  iron  wires ;  by  separating  the  iron  in  this  way 
the  eddy  currents  which  would  be  produced  in  it,  were  it  in 
one  continuous  mass,  are  greatly  reduced. 


Fig.  243. 

Various  arrangements  are  in  use  to  secure  the  interruption 
of  the  primary  ;  the  simplest,  shewn  in  Fig.  243  a,  consists  of 
a  stiff  strip  of  brass  or  iron  which  carries  a  soft-iron  hammer 
at  one  end,  and  is  placed  so  that  the  hammer  in  the  equilibrium 
position  rests  near  the  end  of  the  core.  A  screw  with  a 
platinum  point  presses  against  a  bit  of  platinum  on  the  spring. 
This  screw  is  connected  to  the  battery  while  the  spring  itself 
is  in  connexion  with  one  end  of  the  coil.  The  other  end  of 
the  coil  is  connected  to  the  second  pole  of  the  battery,  and  in 
this  condition  the  circuit  is  complete.  The  soft  iron  core  is 
magnetised  by  the  current  and  attracts  the  hammer  and 
spring.  The  contact  between  the  platinum  points  is  thus 
broken  and  the  current  stopped.  The  core  ceases  to  be 
magnetised  and  the  spring  draws  the  hammer  back  again 


390 


ELECTRICITY 


[CH.  XXIV 


making  contact  between  the  platinum  points  and  completes 
the  circuit. 


Ljr_L 


Fig.  243  a. 

The  condenser  is  connected  between  the  spring  and  the 
screw.  When  the  circuit  is  broken  the  energy  of  the  induced 
current  at  break,  instead  of  producing  a  spark  between  the 
points  and  thus  drawing  out  the  current,  is  employed  in 
charging  the  condenser ;  the  cessation  of  the  current  is  thus 
made  more  rapid.* 

The  secondary  coil  is  wound  on  over  the  primary ;  it 
consists  of  a  very  large  number  of  turns  of  very  fine  wire ; 
with  a  view  of  preventing  too  large  a  potential  difference 
between  adjacent  turns  the  wire  is  wound  in  sections.  By 
means  of  such  an  arrangement  the  electromotive  force  of  the 
secondary  current  is  made  many  times  as  great  as  that  of  the 
primary  and  a  long  spark  can  thus  be  obtained  from  a  battery 
of  a  few  volts. 

The  quantity  of  electricity  in  the  secondary  current 
however  is  not  increased  in  the  same  ratio,  for  we  have 
seen  that  it  is  equal  to  MijR  where  R  is  the  resistance  of  the 
secondary.  Now  R  is  strictly  and  M  approximately  propor- 
tional to  the  number  of  turns  of  the  secondary  so  that  the 
ratio  MjR  is  not  increased  by  putting  more  turns  on  to  the 
secondary. 

An  induction  coil  is  usually  fitted  with  a  commutator  such 
as  has  been  described  in  Section  157,  Fig.  151,  in  order  to 
reverse  the  direction  of  the  primary  current. 


238-239]  ELECT  ROM  AGNETIC    INDUCTION  391 

In  the  induction  coil  a  sudden  break  is  secured  by 
placing  a  condenser  in  parallel  with  the  coil  in  such  a  way 
that  when  the  circuit  is  broken  the  induced  primary  current 
of  self-induction  which  without  the  condenser  would  form  a 
small  arc  across  the  points  at  which  the  break  occurs  is  used 
in  charging  the  condenser  ;  thus  the  cessation  of  the  current 
is  made  more  rapid  than  without  the  condenser,  and  the 
whole  of  the  secondary  flux  is  concentrated  into  a  shorter 
time.  The  secondary  current  is  increased.  Some  recent  experi- 
ments of  Lord  Rayleigh  have  shewn  that  the  same  effect 
can  be  produced  without  the  use  of  a  condenser  by  making 
the  break  sufficiently  sudden  while  if  the  condenser  be  too 
large  its  action  may  reduce  the  secondary  current. 

239.  Transformers.  A  machine  employed  for  the 
purpose  of  changing  the  electromotive  force  of  an  alternating 
current  is  called  a  transformer. 

The  induction  coil  and  the  arrangement  described  in  Section 
234  constitute  transformers;  if  the  number  of  turns  in  the 
secondary  be  m  times  as  great  as  in  the  primary  then  the 
secondary  E.M.F.  is  approximately  m  times  as  great  as  that 
of  the  primary.  In  a  transformer  m  may  be  greater  or  less 
than  unity;  in  the  first  case  as  in  the  induction  coil  tire  E.M.F. 
is  transformed  upwards  by  the  machine;  in  the  latter  it  is 
transformed  downwards;  the  E.M.F.  in  the  secondary  is  less 
than  that  in  the  primary. 

In  order  to  cause  as  many  as  possible  of  the  lines  of 
induction  due  to  the  primary 
circuit  to  pass  through  the 
secondary,  the  iron  circuit  of  a 
transformer  is  usually  com- 
pletely closed,  forming  a  ring- 
round  which  the  two  wire  cir- 
cuits are  wound,  as  shewn  in 
Fig.  244. 

In  electric  light  and  power  j?ig.  214. 

installations    transformers    are 

usually  employed  to  reduce  the  volts  of  a  supply  system  from 
the  value  which  has  been  found  suitable  for  transmission  from 
the  central  station  to  that  suited  to  the  consumers'  lamps 


392  ELECTRICITY  [CH.  XXIV 

or  motors.  The  consumer  needs  a  certain  supply  of  energy. 
The  stfpply  of  this  per  unit  of  time  is  measured  by  the 
product  EG  of  the  volts  and  amperes.  To  supply  a  large 
current  however  requires  a  large  amount  of  copper  in  the 
leads  otherwise  the  loss  due  to  heating  becomes  too  great ; 
it  is  therefore  in  many  cases  economical  to  supply  a  small 
current  at  a  high  voltage.  The  consumer  may  require  for 
his  work  a  larger  current  but  at  a  less  voltage ;  this  can 
be  given  him  by  means  of  a  step-down  transformer. 

In  some  cases  two  transformers  may  be  employed,  e.g. 
in  working  a  large  tramcar  system  or  on  a  railway.  The 
machine  produces  the  electricity  at  a  moderate  voltage.  It  is 
transformed  up  in  order  that  copper  may  be  economised  in 
the  leads  which  carry  it  to  the  more  distant  parts  of  the 
S}^stem.  At  these  distant  points  it  is  transformed  down  so 
as  to  be  conveniently  supplied  to  the  motors  of  the  cars. 

240.  Electromagnetic  Machines.  We  have  seen 
in  Section  234  how  a  supply  of  electricity  can  be  obtained  by 
suitably  rotating  a  circuit  in  a  magnetic  field.  The  experiment 
there  described  contains  the  germ  of  the  many  dynamos  and 
electromagnetic  generators  in  existence.  In  its  simplest  form 
such  a  machine  would  consist  of  a  single  elongated  coil  of  a 
few  turns  mounted  as  shewn  in  Fig.  245  a  between  the  poles 
of  a  magnet.  This  coil  can  be  made  to  rotate  about  a  hori- 
zontal axis  lying  in  the  plane  of  the  paper;  its  ends  are 
connected  to  a  split  ring  commutator  giving  direct  current  in 


Fig.  245  a.  Fig.  245  6. 

the  external  circuit  or,  for  an  alternate  current  machine,  to 
two  slip  rings,  245  b.  The  rotating  coil  is  known  as  the 
armature  of  the  machine.  As  the  coil  rotates  the  E.M.F.  changes 


239-241]  ELECTROMAGNETIC    INDUCTION 


393 


from  zero  when  the  plane  of  the  coil  is  perpendicular  to  the 
lines  of  force  to  a  maximum  when  it  is  parallel  to  them. 

241.       Magneto-electric     Machines.       The    small 
magneto-electric    machines,     Fig.     246,     used    sometimes    in 


Fig.  246. 

medical  practice  act  on  the  same  principle.  Two  coils  of  wire 
with  parallel  iron  cores  are  connected  in  series  and  arranged 
so  that  they  can  rotate  about  a  common  axis  parallel  to  the 
cores.  The  rotation  takes  place  in  front  of  a  horseshoe  magnet 
so  that  the  ends  of  the  cores  come  alternately  opposite  to  the 
poles  of  this  magnet.  The  other  extremities  of  the  cores  are 


394 


ELECTRICITY 


[CH.  XXIV 


connected  by  a  bar  of  soft  iron.  Thus  in  one  position  of  the 
coils  lines  of  induction  run  from  the  north  pole  of  the 
magnet  through  the  axis  of  one  coil  and  return  to  the 
south  pole  of  the  magnet  through  the  axis  of  the  other ; 
on  turning  the  coils  through  180°  the  direction  in  which 
the  lines  of  induction  are  linked  with  them  is  reversed. 
Thus  an  electromotive  force  is  produced  round  the  coils  which 
is  reversed  at  each  half  rotation.  The  ends  of  the  coils  come 
down  to  slip  rings  against  which  brushes  rub,  arid  an  alter- 
nating current  is  produced  in  an  external  circuit  connected 
with  the  brushes.  By  using  a  commutator  instead  of  two 
slip  rings  a  direct  current  is  obtained. 

242.     Shuttle  wound  Armature.     Returning  now  to 
the  single  coil  described  in  Section  240,  imagine  a  second  coil 
to  be  fixed,  as  in   Fig.   247,  on  the 
same  shaft  in  a  plane  perpendicular  to 
the  first  and  let  this  commutator  be 
divided  into  four  segments — two  for 
each    coil  —  instead    of     into    two. 
Further  let   the  coils    be    connected  Fig.  247. 

to  the  segments  so  that  each  in  turn 

is  connected  to  the  brushes  when  near  the  position  for  a 
maximum  electromotive  force.  The  fluctuations  of  the  voltage 
in  the  external  circuit  will  be  reduced  and  the  average  voltage 
raised. 

By  continuing  the  process  and  increasing  still  further  the 
number  of  the  sections  we 
arrive  at  the  Siemens  shuttle 
wound  armature  in  which  a 
large  number  of  coils  are 
wound  on  a  central  boss  of 
laminated1  soft  iron,  each 
coil  being  connected  to  two 
sections  of  the  commutator, 
the  brushes  of  which  are 
arranged  so  as  to  tap  off  the 


current  from  each  section  in 
turn    as    the    E.M.F.    in   that 


Fig.  248. 


1  The  iron  is  made  of  a  number  of  thin  laminse  to  reduce  eddy  currents. 


241-243]  ELECTROMAGNETIC   INDUCTION  395 

section  reaches  its  maximum  value.  Tn  each  case  the  end  of 
one  coil  and  the  beginning  of  the  next  are  connected  to  the 
same  section  of  the  commutator  so  that  the  various  coils  on 
the  armature  form  a  continuous  circuit.  Fig.  248  shews  such 
an  armature. 

This  armature  rotates  between  the  poles  of  a  magnet 
shaped  so  as  to  come  as  close  as  possible  to  the  wire  of 
the  coils  and  thus  increase  the  induction  through  them. 

243.  Dynamo  Machines.  To  induce  the  current  a 
permanent  magnet  may  be  used,  but  in  modern  machines 
an  electromagnet  is  ordinarily  employed.  The  coils  of  this 
magnet  are  spoken  of  as  the  field  coils  of  the  machine.  The 
field  coils  may  be  supplied  with  current  from  some  external 
source;  the  machine  is  then  said  to  be  separately  excited. 

Separate  excitation  however  is  not  always  required,  for 
suppose  the  field  coils  are  connected  up  in  series  with  the 
armature  through  the  brushes  and  the  machine  started. 
Sufficient  traces  of  magnetisation  are  usually  left  in  the  iron 
of  the  machine  to  start  a  small  current ;  this  passing  round 
the  field  coils  magnetises  them  more  strongly,  thus  increasing 
the  current  which  again  reacts  on  the  field,  the  machine  is 
then  self-excited.  Such  a  machine  is  spoken  of  as  a  Dynamo 
Machine. 

The  volts  produced  by  a  dynamo  depend  on  its  construction 
and  on  the  nature  of  the  iron  used  in  it,  they  also  depend 
on  the  speed  being,  when  the  magnetisation  of  the  iron 
approaches  saturation,  approximately  proportional  to  the  speed. 
In  this  case  the  strength  of  the  field  is  nearly  constant,  and 
the  number  of  lines  of  induction  cut  in  a  given  time  by  the 
armature  will  be  proportional  to  the  speed. 

For  some  purposes  the  coils  of  a  dynamo  are  connected  so 
that  the  whole  current  through  the  armature  passes  through 
the  external  circuit,  and  the  field  coils  in  series.  The  machine 
is  said  to  be  series  wound,  Fig.  249.  Since  the  whole  current 
has  to  traverse  the  field  coils,  these  consist  of  a  few  turns  of 
thick  wire. 

In  this  case  if  the  resistance  in  the  external  circuit 
decreases  the  current  in  the  armature  and  field  coils  increases. 


396 


ELECTRICITY 


[CH.  XXIV 


The  increase  of  the  current  in  the  field  coils  produces  for 
constant  speed  a  rise  in  the  E.M.F.  of  the  machine  which  tends 
to  raise  the  external  current  still  further,  thus  the  machine  is 
not  suitable  for  steady  working  in  a  circuit  of  variable  resis- 
tance, e.g.  in  an  incandescent  lamp  circuit  where  the  resistance 
depends  on  the  number  of  lamps  in  use. 


Fig.  249. 


Fig.  250. 


In  a  shunt  wound  machine,  Fig.  250,  the  field  coils  consist 
of  a  large  number  of  turns  of  thin  wire,  the  ends  of  which 
are  connected  to  the  brushes.  Thus  the  field  coils  and  the 
external  circuit  are  in  parallel. 

If  for  such  a  machine  the  resistance  in  the  external  circuit 
falls,  thus  causing  the  machine  for  a  given  voltage  to  give  an 
increased  current,  a  smaller  fraction  of  that  greater  current 
circulates  round  the  field  coils ;  thus  the  strength  of  the  field 
does  not  necessarily  increase  as  the  current  in  the  armature 
increases  and  the  voltage  of  the  machine  remains  more  nearly 
constant  for  varying  currents  than  is  the  case  in  a  series 
machine. 


243-244]  ELECTROMAGNETIC   INDUCTION 


397 


The  rise  or  fall  in  voltage  depends  on  the  relation  between 
the  various  resistances  which  will  determine  whether  the  rise 
produced  in  the  current  in  the 
field  coils  is  greater  or  less  than 
the  fall  due  to  the  fact  that  a 
smaller  fraction  of  that  current 
circulates  through  them. 

In  a  compound  machine,  Fig. 
251,  some  of  the  field  coils  are  in 
series  with  the  external  circuit  as 
in  a  series  machine,  and  in  addition 
to  this  a  number  of  turns  of  thin 
wire  are  wound  on  in  parallel  with 
the  external  circuit. 

When  the  external  resistance 
is  reduced  the  current  in  the 
armature  rises;  so  far  as  the  series 
coils  are  concerned  the  field  in- 
creases; a  less  fraction,  however, 
of  the  whole  current  circulates 
through  the  shunt  coils  and  the 
resistances  are  arranged  (1)  so  that 


Fig.  251. 


the  field  in  consequence  is  reduced,  and  (2)  so  that  the  reduc- 
tion just  counterbalances  the  rise  due  to  the  action  of  the 
series  coils :  thus  the  voltage  remains  constant. 

244.  The  Gramme  ring.  Another  form  of  armature 
is  the  Gramme  ring.  Let  us  suppose  we  have  a  field  of 
magnetic  force  in  which  the  lines  are  parallel  to  the  paper,  as 
shewn  in  Fig.  252.  Consider  a  coil  whose  plane  is  perpendi- 
cular to  the  paper,  and  let  it  move  about  an  axis  parallel  to 
its  own  plane  and  perpendicular  to  the  paper.  When  the  coil 
is  in  the  position  A,  a  maximum  number  of  lines  of  force  pass 
through  it.  As  it  rotates  into  position  J2,  the  number  de- 
creases until  none  pass  through ;  from  B  to  C  the  decrease 
continues,  the  lines  of  force  again  become  linked  with  the  coil, 
but  in  the  negative  direction  opposite  to  that  in  which  they 
previously  passed  through,  and  at  C  the  number  of  negative 
lines  is  a  maximum.  Thus  during  the  motion  from  A  to  C 
an  E.M.F.  acts  on  the  coil  and  a  current  circulates  in  it.  After 


398  ELECTRICITY  [CH.  XXIV 

passing  C  the  negative  lines  decrease  in  number  to  vanish 
at  D  when  lines  begin  to  traverse  the  coil  in  the  positive 
direction;  thus  from  C  to  A  an  E.M.F.  acts  round  the  coil 
in  the  opposite  direction  to  that  previously  existing.  If 
the  ends  of  the  coil  be  carried  to  two  rings  on  the  axis  and 


Fig.  252. 

brushes,  connected  with  the  external  circuit,  rub  against  these 
an  alternating  current  is  produced  in  this  circuit;  by  using  a 
commutator  instead  of  the  rings,  the  current  may  be  made  to 
traverse  the  external  circuit  always  in  the  same  direction. 
The  E.M.F.  in  the  circuit  varies  as  the  coil  rotates,  and  is 
greatest  when  the  coil  is  in  position  B  or  D, 

If  now,  instead  of  a  single  coil  we  have  a  series  of  such 
coils  arranged  so  that  their  centres  lie  on  a  circle  with  ite 
centre  in  the  axis  of  rotation,  and  corresponding  to  each  have 
two  segments  of  a  commutator  set  so  as  to  pick  off  the  current 
from  each  coil  when  the  E.M.F.  in  that  coil  is  a  maximum,  we 
increase  the  current  greatly. 

If  further  the  coils  be  all  wound  about  a  core  of  iron  wires 
shaped  like  an  anchor  ring  the  magnetic  induction  and  the 
corresponding  E.M.F.  will  be  greatly  increased. 

Moreover  the  coils  may  be  all  continuous  as  in  the 
diagrammatic  illustration  of  Fig..  253. 

A  continuous  length  of  wire  is  wound  round  the  ring- 
shaped  core  and  from  a  number  of  equidistant  points  round 
the  ring  wires  are  led  to  the  respective  sections  of  the 
commutator. 


244-245]  ELECTROMAGNETIC    INDUCTION 


399 


Brushes  make  contact  at  E  and  E'  respectively. 

Owing  to  electromagnetic  induction  all  the  turns  of  the 
armature  in  the  half  ABC  tend  to  produce  a  current  traversing 
the  armature  in  the  direction  ABC,  those  in  the  half  CD  A 
tend  to  produce  a  current  in  direction  ADC.  These  two 
currents,  each  of  strength  7/2  let  us  suppose,  combine  to 
produce  in  the  external  circuit  a  current  of  strength  /. 


Fig.  253. 

We  notice  that  since  the  voltage  of  the  machine  between 
K  and  E'  is  that  due  to  all  the  turns  on  the  half  ABC  it  may 
be  considerable;  the  resistance,  however,  of  the  armature, 
being  that  due  to  all  the  turns,  is  also  considerable ;  hence  the 
machine  is  not  suited  to  give  a  very  large  current. 

As  in  the  shuttle  wound  form  of  the  dynamo  the  field 
magnets  are  usually  electromagnets,  the  current  being  started 
by  the  residual  magnetism  of  the  iron,  the  machine  being 
either  series,  shunt  or  compound  wound.  In  the  figure  a  series 
wound  machine  is  shewn. 

245.  Armature  Reactions.  It  should  be  noted  that 
the  current  in  the  armature  produces  its  own  field  of  magnetic 
force,  hence  the  lines  of  induction  through  the  armature  are 


400 


ELECTRICITY 


[CH.  XXIV 


not  solely  those  due  to  the  field  magnets,  and  the  theory  is 
more  complex  than  that  given  above;  besides,  since  the  current 
round  the  armature  varies  continually,  effects  of  self-induction 
have  to  be  considered ;  in  consequence,  in  an  actual  machine 
when  working,  the  brushes  have  not  the  position  shewn  but 
are  displaced  in  the  direction  of  motion. 

246.  Alternate  Current  Machines.  Modern  alter- 
nate current  machines  are  more  complex  than  that  figured  in 
Fig.  245  b — a  simple  coil  connected  to  two  slip  rings.  Thus  the 
field  magnets  of  a  Siemens  machine  consist  of  two  sets  of 
electromagnets.  The  ends  of  the  axes  of  each  set  are  ar- 
ranged in  a  circle,  the  axes  themselves  being  perpendicular  to 
the  plane  of  the  circle  and  the  poles  being  alternately  positive 
and  negative.  The  two  series  of  poles  are  placed  with  their 
planes  parallel  and  at  a  short  distance  apart  in  such  a  position 
that  the  north  poles  of  one  set  are  opposite  to  the  south  poles 
of  the  other  and  vice  versd,  as  in  Fig.  254.  Thus  between  the 


Fig.  254. 


two  there  is  a  field  of  force  ;  arid  the  lines  of  force  will  run 
alternately  upwards  and  downwards. 

The  armature  revolves  in  the  gap  between  the  two  series  of 


245-247]  ELECTROMAGNETIC    INDUCTION  401 

magnets.  It  consists  of  a  series  of  coils,  the  same  in  number 
as  the  magnets,  placed  round  the  circumference  of  a  disc, 
with  their  axes  parallel  to  those  of  the  magnets,  in  such  a  way 
that  their  centres  are  equally  distributed  round  a  circle  of  the 
same  diameter  as  that  on  which  the  poles  of  the  magnets  lie. 

The  windings  of  the  alternate  coils  are  opposite  in  direction, 
the  wire  is  continuous  round  all  the  coils  and  its  two  ends  are 
connected  to  slip  rings  on  the  axle.  From  these  the  current 
is  conveyed  by  means  of  brushes  to  the  external  circuit. 
Now  start  from  a  position  in  which  the  centres  of  the  armature 
coils  lie  on  the  axes  of  the  magnets.  In  one-half  the  coils, 
as  the  machine  rotates,  the  number  of  lines  of  force  which 
pass  through  from  below  upwards  is  increased,  in  the  other 
half  it  is  decreased.  Thus  the  electromotive  force  round  the 
alternate  coils  of  the  armature  is  opposite ;  but  the  direction  of 
the  windings  round  alternate  coils  is  also  opposite,  hence  the 
electromotive  force  retains  everywhere  the  same  direction 
through  the  wire  of  the  armature.  When  the  armature  is 
moved  the  E.M.F.  starts  from  zero,  gradually  rises  and  falls  again 
to  zero  as  the  armature  coils  again  come  opposite  to  the  magnet 
coils.  As  the  motion  continues  the  same  process  is  repeated, 
but  the  E.M.F.,  as  the  armature  coils  move  one  section  forward, 
is  opposite  in  direction  to  that  in  the  previous  section;  an 
alternating  current  is  produced,  it  passes  through  the  value 
zero  each  time  the  moving  poles  of  the  armature  come  opposite 
the  fixed  poles  of  the  magnets  and  has  opposite  signs  between 
each  consecutive  pair  of  such  positions. 

There  are  various  other  forms  of  alternate  current  machines,  for 
which  reference  should  be  had  to  special  books  on  the  subject. 

See  for  example  S.  P.  Thompson's  Dynamo  Electric  Machinery. 

In  some  of  these  the  armature  is  fixed  while  the  field  magnets  rotate, 
the  advantage  of  this  is  that  a  commutator  is  not  then  required  in  a 
moving  part  of  the  circuit  in  which  the  alternate  current  is  circulating. 

It  should  be  noted,  moreover,  that  the  current  does  not  alternate  in 
the  field  magnets ;  in  some  cases  permanent  magnets  are  used,  in  others 
a  small  direct  current  machine  is  employed  to  excite  them. 

247.  Electric  Motors.  In  a  dynamo  machine  me- 
chanical energy  is  transformed  into  electrical.  The  armature 
is  made  to  rotate  in  a  magnetic  field  and  a  current  is  formed. 
This  action  is  in  all  cases,  with  direct  current  machines, 


G.  E. 


26 


402 


ELECTRICITY 


[CH.  XXIV 


reversible ;  if  we  supply  a  current  the  armature  will  rotate,  the 
dynamo  becomes  a  motor ;  electrical  energy  can  by  its  aid  be 
transformed  into  mechanical.  There  will  be  as  many  forms  of 
motors  as  there  are  of  dynamos. 

When  a  given  E.M.F.  is  first  applied  to  the  terminals  of  a 
motor  at  rest  the  resistance  is  small,  a  large  current  passes 
and  the  machine  begins  to  rotate.  This  rotation  sets  up  an 
E.M.P.  in  the  machine  opposite  to  that  driving  it,  and  the 
current  is  reduced;  if  the  driven  machine  is  doing  no  w"ork 
and  if  there  be  no  friction  the  speed  will  go  on  increasing 
until  this  opposing  E.M.F.  is  equal  to  that  applied  to  the 
machine ;  the  current  then  vanishes. 

248.     Transformations  of  Energy   in   a   Motor. 

Now  let  us  call  E  the  impressed  E.M.F.  and  e  the  back  E.M-.F.  of 
the  motor.    If  the  machine  is  doing  work  the  back  E.M.F.  of  the 
motor  is  less  than  the  impressed  E.M.F.  and   the   current    is 
(E  —  e)/£,  where  R  is  the  resistance  of  the  circuit.     If  we  call 
this    current    7,    the    energy  supplied  per  unit   time  to    the 
machine  from  outside  is  El,  the  energy  used  by  the  motor  in 
doing  external  work  is  el,  and  the  energy  lost  as  heat  in  the 
armature  is  (E  —  e)  7,  or  writing  in  the  value  of  /  we  have 
Energy  supplied  to  system  E(E-e}jR, 
Energy  transformed  by  motor  e(E-e)jR, 
Energy  lost  as  heat  (E-efjR. 

We  can  represent  the  first  two  of  these?  quantities  in  a 
diagram  thus,  Fig.  255. 

Let  ABCD  be  a  square  of  which 
the  side  A  B  represents  E.  • 

Take  AK  and  AM  each  equal  to 
e,  and  draw  MON  and  LOK  parallel 
to  AB  and  BC. 

Then  KB  and  MD  are  each 
equal  to  JE  —  e,  and  the  parallelo- 
gram LB  is  equal  to  E(E-e)  and 
therefore  represents  the  energ}r  sup-  Fig.  255. 

plied,   while   the  parallelogram    NK 
is  e(E-e)  and   represents  the  work  done  Hy  the   motor. 


247-249]  ELECTROMAGNETIC    INDUCTION  403 

As  e  increases,  that  is  as  K  moves  along  towards  B,  the 
area  of  this  parallelogram  increases  at  first  and  then  decreases, 
reaching  its  maximum  value  when  K  is  midway  between  A  and 
B  so  that  e  is  equal  to  Ej'2.  In  this  case  the  external  work 
done  is  a  maximum. 

Thus  a  motor  does  the  greatest  amount  of  work  when  its 
speed  is  such  that  the  back  E.M.F.  is  half  the  impressed.  The 
current  then  is  |  EjR,  it  has  half  the  value  it  would  have  if 
the  motor  were  at  rest. 

By  the  efficiency  of  a  motor  is  meant  the  ratio  of  the  work 
it  does  to  the  energy  supplied.  Thus  the  efficiency  is  measured 
by  e(E-e)jE(E-e)  or  e/E ;  it  is  greatest  then  if  e  =  E,  In 
this  case  the  motor  takes  no  energy  from  the  source  and  does 
no  work.  A  distinction  then  has  to  be  drawn  between  the 
condition  for  maximum  efficiency  and  that  for  maximum  work 
done ;  in  the  latter  case,  since  e  =  ^E  the  efficiency  is  only 
one-half. 

Since  the  energy  lost  as  heat  is  (E  —  e)2/R  we  notice  that 
when  the  external  work  is  a  maximum  and  e  is  \K  its  value  is 
^E*IR.  Under  these  conditions  the  external  work  e(H—e)jR 
is  also  \  E*jR,  while  the  total  energy  supplied  is  J  E2jR.  Thus 
when  the  external  work  is  a  maximum  it  is  equal  to  half  the 
energy  supplied. 

249.  Starting  a  Motor.  The  back  electromotive 
force  in  a  motor  does  not  reach  its  full  value  until  the  motor 
has  acquired  its  final  speed ;  accordingly  the  current  taken  by 
a  motor  at  starting  is  greater  than  that  which  it  ultimately 
requires  to  drive  it ;  indeed,  if  the  full  voltage  were  applied 
direct  to  the  armature  the  current  would  be  too  great,  and  the 
armature  would  be  destroyed.  For  this  reason  a  starting 
resistance  is  usually  employed  in  series  with  the  armature  of 
the  motor  ;  the  volts  are  applied  at  first  through  this  resistance 
which  is  gradually  cut  out  of  the  circuit  as  the  speed  rises, 
until  finally  the  full  pressure  of  the  supply  is  on  the  terminals 
of  the  machine. 


26—2 


404  ELECTRICITY  [CH.  XXIV 

EXAMPLES  ON  ELECTROMAGNETISM   AND 
ELECTROMAGNETIC  INDUCTION. 

1.  A  single  turn  of  wire  coiled  into  the  form  of  a  circle  of  12  cm. 
radius  is  placed  in  the  magnetic  meridian,  and  two  circular  turns  of  wire 
of  radius  24  cm.  are  placed  so  that  their  common   plane   and   centre 
coincide  with  that  of  the  first  circle.     A   small   magnet   is   suspended 
horizontally  at  the  centre.     What  effect,  if  any,  will  be  produced  on  the 
magnet  if  the  same  current  be  sent  in  one  direction  through  the  single 
wire,  and  in  the  other  direction  through  the  double  wire? 

2.  The  single  wire  mentioned  above  (see  Question1 1)  is  turned  about 
a  vertical  axis  until  in  a  plane  perpendicular  to  its  former  position,  and 
the  new  position  of  the  needle  is  observed.     The  direction  of  the  current 
in  the  single  wire  is  then  reversed ;  calculate  the  deflexion  of  the  needle 
produced  by  the  reversal  from  the  following  data : 

Current  flowing  through  the  coils  =  10  amperes, 

H  =  *18  C.G.  s.  unit. 

3.  A  current  of  10  amperes  flows  along  an  indefinitely  long  straight 
wire ;  find  the  force  which  it  will  exert  upon  a  magnetic  pole  of  strength 
20  units  placed  at  a  distance  of  (3  cm.  from  the  wire. 

4.  A  wire  is  bent  into  the  form  of  a  circle  of  10  cm.  radius,  and  a 
current  of  2  amperes  is  passed  through  it.     Find  the  force  exerted  by 
the  current  at  a  point  on  the  axis  and  at  a  distance  of  20  cm.  from  the 
circumference. 

5.  Two  infinitely  long   straight   wires   are  placed  parallel  to  one 
another,   and   currents    of    5   and  8   amperes   respectively  are  passed 
through  them  in  the  same  direction.     Find  (1)  the  magnitude,  (2)  the 
direction  of  the  force  acting  upon  unit  length  of  either  wire. 

6.  A  platinum  wire  is  hung  from  a  loop  in  a  copper  wire  so  that  its 
lower  end  just  dips  into  a  vessel  containing  mercury,  and  the  copper 
wire  and  the  mercury  are  connected  to  the  opposite  poles  of  a  battery. 
Describe  and  explain  the  movements  that  will  take  place. 

7.  A  large   circular  coil   is   in   series  with    a    smaller   one   and   a 
ballistic  galvanometer.     When  the  larger  coil  is  quickly  turned  through 
180°  about  a  vertical  axis  in  the  Earth's  field,  a  deflexion  of  30  divisions 
results.     The  smaller  coil  when  simiiarly  turned  between  the  poles  of  a 
horse-shoe  magnet  produces  a  deflexion  of  70  divisions.     Calculate  the 
value  of  the  field  in  which  the  smaller  coil  is  turned.     The  following  are 
the  necessary  data : 

Number  of  turns  on  large  coil  =  80. 

,,          ,,         ,,       small  ,,   =10. 

Radius  of  large  coil  =  50  cms. 

„      ,,    small  ,,   =   4    „ 

H=-IS  c.o.s.  units. 


EXAMPLES  405 

8.  Two  horizontal  brass  rods  are  placed  parallel  and  at  a  distance  of 
1  metre  apart.     A  third  rod  slides  over  them  parallel  to  itself  with  a 
uniform  velocity  of   10  metres   per  second.      Find  in  volts  the  E.M.F. 
between  the  ends  of  the  fixed  rods — assuming  the  earth's  vertical  magnetic 
force  to  be  -47  c.  G.S.  unit. 

9.  A  hoop  of  copper  is  set  rotating  about  a  diameter  as  an  axis. 
It  is  placed  in  a  magnetic  field  with  its  axis  of  rotation  (1)  parallel, 
(2)  perpendicular  to  lines  of  magnetic  force.     If  the  mechanical  friction 
and  the  initial  velocity  of  rotation  are  in  both  cases  the  same,  explain 
why  it  comes  to  rest  in  a  shorter  time  in  one  position  than  when  in  the 
other. 

10.  A  bar  of  soft  iron  is  thrust  into  the  interior  of  a  coil  of  wire, 
whose  terminals  are  connected  to  a  galvanometer.     Could  the  coil  and 
bar  be  placed  in  such  a  position  that  no  induced  current  might  pass 
through  the  galvanometer? 

11.  The  poles  of  a  battery  are  connected  in  turn  by 

(1)  a  long  straight  insulated  wire, 

(2)  the  same  wire  coiled  into  a  close  spiral, 

(3)  ,,        ,,        ,,        ,,       round  a  soft  iron  core. 

Describe  and   discuss  what  happens  in  each  case  on  breaking  the 
circuit. 


CHAPTER  XXV. 


TELEGRAPHY  AND  TELEPHONY. 


250.  The  Electric  Telegraph.  In  the  electric 
telegraph  the  current  is  used  to  transmit  a  signal  to  a 
distance.  In  its  simplest  form  a  wire  uniting  the  two  places 
and  carried  on  insulating  supports  is  connected  at  the  trans- 
mitting end,  A,  Fig.  256,  through  a  key  with  one  pole  of  a 


Fig.  256. 

battery;  the  other  pole  of  the  battery  is  put  to  earth 
by  being  connected  to  a  large  metal  plate  sunk  in  the  ground. 
At  the  distant  station,  £,  the  wire  is  joined  to  one  terminal  of 
a  galvanometer  of  which  the  second  terminal  is  put  to  earth 
in  the  same  manner.  On  depressing  the  key  a  current 
passes  through  the  wire  deflecting  the  galvanometer  needle,  and 
returns  by  the  earth  to  the  sending  station,  and  thus  signals 


250-251]  TELEGRAPHY    AND   TELEPHONY 


407 


can  be  transmitted  from  A  to  B.  The  needle  of  the  galva- 
nometer usually  swings  in  a  vertical  plane,  its  motions  are 
indicated  by  a  pointer  outside  the  case  of  the  instrument. 
By  means  of  a  double  key  at  A  either  end  of  the  battery 
can  be  connected  at  will  to  the  line;  thus  the  direction  of  the 
currents  and  therefore  the  direction  of  the  deflexion  in  the 
receiving  instrument  can  be  reversed.  •  Each  letter  of  the 
alphabet  is  represented  by  a  suitable  combination  of  deflexions 
to  right  and  left. 

By  adding  a  second  receiver  at  A  and  a  second  battery  and 
key  at  £,  connected  as  shewn  in  Fig.  256,  the  above  line  can 
be  made  to  work  either  way.  With  the  keys  in  their  normal 
positions  both  ends  of  the  lines  are  earthed,  and  as  shewn 
in  the  figure  the  negative  poles  of  both  batteries  are  to  earth, 
the  positives  being  insulated.  When  the  key  A  is  depressed 
the  current  passes  and  is  registered  on  the  receiver  at  B. 


Fig.  257. 

251.  The  Morse  Instrument.  In  the  Morse  re- 
ceiver, Fig.  257,  a  narrow  continuous  strip  of  paper  is  drawn 
by  clockwork  from  a  drum  between  rollers. 

A  lever,  AB,  with   a  pen  or  style  at  the  end  B  carries 


408  ELECTRICITY  [CH.  XXV 

a  piece  of  soft  iron  at  A  :  the  current  from  the  distant  station 
can  pass  round  an  electromagnet  below  A  ;  when  this  is  the 
case  the  iron  is  attracted  and  the  pen  drawn  up  against  the 
moving  paper  ;  when  this  circuit  is  broken  a  spring  draws  the 
pen  away  again  ;  thus  a  dot  or  a  dash  is  marked  on  the  paper 
depending  upon  the  length  of  time  during  which  the  contact 
is  maintained. 

The  letters  are  represented  by  a  combination  of  dots 
or  short  strokes  and  dashes  or  long  strokes  as  shewn  in  the 
next  section. 

In  practice  the  current  from  the  sending  station  would 
probably  not  be  strong  enough  to  move  the  lever  of  the  Morse 
instrument,  it  is  therefore  used  to  work  a  relay  instead. 

This  instrument  consists  of  a  very  light  lever  actuated  by 
an  electromagnet  through  which  the  distant  current  passes. 
When  the  lever  is  attracted  by  the  electromagnet  it  closes 
the  circuit  of  a  local  battery  through  the  Morse  instrument. 
Since  this  current  has  only  to  traverse  the  coils  of  the  instru- 
ment and  the  wires  connecting  them,  instead  of  the  many 
miles  of  the  line,  it  can  easily  be  made  of  sufficient  strength 
to  work  the  instrument. 

The  Morse  key  which  is  used  as  a  sending  instrument  is 
shewn  in  Fig.  170,  the  line  is  connected  to  the  fulcrum  of  the 
lever,  and  in  the  normal  position  of  the  key  is  in  circuit  with 
the  relay;  on  depressing  the  fulcrum  this  circuit  is  broken 
and  connexion  is  made  between  the  line  and  the  battery. 


U 
V 
W 
X 
Y 


253.     The  Telephone.     In  a  telephone   use  is  made 
of  the  currents  produced  when  a  piece  of  magnetised  soft  iron 


252 

.     The  Morse  Alphabet. 

A 

,  K     •  

B 

_  .  .  .                     L      •  —  .  . 

C 

_  .  _  .                  M     

D 

_  .  .                        N      —  • 

E 

0     

F 

.  .                    p         . 

G 

Q      

H 

...                          R      .  -   . 

I 

s    ... 

J 

T     — 

251-254]  TELEGRAPHY   AND   TELEPHONY 


409 


in  the  form  of  a  thin  diaphragm  is  made  to  vibrate  in  front 
of  a  coil  of  wire. 

Bell's  telephone  used  originally  as  a  receiver  and  a  trans- 
mitter took    the  form   shewn   in  Fig.    258.      A  coil   of    wire 


Fig.  258. 

is  wound  on  a  flat  bobbin  which  surrounds  one  pole  of  a 
permanent  magnet,  the  ends  of  the  coil  come  to  binding 
screws  connected  to  the  external  circuit.  A  disc  of  thin 
sheet-iron  is  supported  at  its  edges  very  close  to  the  pole 
of  the  magnet  which  carries  the  coil.  This  constitutes  the 
transmitter.  At  the  distant  station  a  similar  apparatus  is 
connected  to  the  circuit.  On  speaking  into  the  transmitting 
instrument  the  magnetised  disc  is  set  into  vibration,  thus 
producing  induction  currents  in  the  coil ;  these  traverse  the 
coil  of  the  distant  instrument  or  receiver,  thus  setting  it  into 
a  similar  state  of  vibration  and  causing  it  to  emit  the  same 
sounds  as  those  to  which  the  motion  of  the  transmitter 
was  due. 


254.  The  Microphone.  The  currents  obtained  from 
Bell's  instrument  used  as  a  transmitter  are  very  weak,  and 
hence  for  this  purpose  it  has  been  displaced  by  instruments 
working  on  the  principle  of  the  microphone. 

If  in  a  circuit  containing  a  battery  and  a  telephone  there 
be  a  loose  contact,  as  in  Fig.  259,  in  which  the  vertical  rod 
rests  loosely  in  its  supports,  and  this  contact  be  set  into 
vibration  by  any  means,  the  variations  of  the  resistance  at  the 
contact  are  so  great  that  corresponding  vibrations  are  pro- 
duced in  the  telephone  and  a  sound  is  emitted ;  the  vibrations 


410 


ELECTRICITY 


[CH.  XXV 


thus  produced  may  exceed  in  amplitude  those  of  the  original 
sound  and  hence  it  may  be  intensified. 


\ 


Fig.  259. 


Fig.  260. 


Various  forms  of  transmitter  based  on  this  principle  are 
in  use  ;  in  the  Blake  instrument,  Fig.  260,  a  platinum  point, 
which  is  mounted  on  a  disc  of  elastic  material,  presses  lightly 
on  a  piece  of  carbon ;  the  current  from  a  battery,  usually  a 
few  Leclanche  cells,  passes  through  this  contact  and  round 
a  Bell  telephone  at  the  receiving  station.  On  speaking  into 
the  transmitter  the  vibrations  of  the  disc  cause  variations  in 
the  resistance  at  the  point  and  hence  in  the  current.  These 
set  the  receiving  disc  into  vibration  and  the  words  are 
reproduced. 

When  the  receiver  is  on  its  hook  the  current  through  the 
telephone  is  broken,  the  line  wire  is  connected  to  an  electric 
bell  at  the  receiving  station ;  this  bell  can  be  rung  by 
pressing  a  button  at  the  transmitter.  On  taking  the  telephone 
off  its  hook  a  spring  causes  the  hook  to  rise,  thus  breaking  the 
bell  circuit  and  closing  that  of  the  telephone. 


CHAPTER   XXVI. 

ELECTRIC   WAVES. 


255.  Electric  Inertia.  In  dealing  with  the  motion 
of  matter1  we  considered  at  some  length  the  phenomena 
©f  inertia,  that  property  of  matter  which  makes  it  persevere 
in  the  state  of  rest  or  motion  in  which  it  finds  itself,  so  long 
as  it  is  not  acted  on  from  without. 

The  electric  current  also  apparently  possesses  this  property 
of  inertia.  We  have  seen  that  when  electromotive  force  is 
applied  to  a  circuit  the  current  does  not  at  once  reach  its 
final  value,  there  is  at  the  start  an  induced  opposing  current 
of  self-induction,  while  when  the  electromotive  force  is 
removed  the  current  of  self-induction  at  break  continues  to 
flow  for  a  brief  period.  We  may  compare  this  to  the  motion 
of  a  body  which  is  resisted  by  some  kind  of  frictional  force 
depending  on  the  velocity. 

When  force  is  applied  to  such  a  body  it  begins  to  move, 
at  first  slowly  in  consequence  of  its  inertia,  then  more  rapidly 
until  the  frictional  force  is  sufficient  to  balance  the  impressed 
force,  when  a  state  of  uniform  motion  is  reached  which 
continues  as  long  as  the  force  acts ;  if  the  force  ceases  to 
act  the  body  continues  to  move  for  a  time  in  consequence 
of  its  inertia,  but  is  shortly  brought  to  rest  by  the  friction. 

We  might  look  upon  the  body  as  moving  all  the  time 
with  constant  velocity  so  long  as  the  external  force  acts, 
balancing  the  friction,  but  as  having  superimposed  on  this,  at 
the  start,  a  velocity  in  the  opposite  direction,  arid  at  the  finish 

1  Dynamics,  Section  77. 


412  ELECTRICITY  [CH.  XXVI 

a  velocity  in  the  same  direction  as  that  of  motion ;  both  these 
die  away,  both  arise  from  the  inertia  of  the  body.  At  the 
start  the  force  is  storing  up  kinetic  energy  in  the  body,  at 
the  finish  this  kinetic  energy  carries  the  body  on,  and  enables 
it  to  do  work  against  friction. 

In  the  same  way  we  may  look  upon  the  effects  of  self- 
induction  in  an  electric  circuit  as  inertia  effects.  When 
the  electromotive  force  is  applied  to  the  circuit  it  takes  time 
to  overcome  the  electric  inertia  of  the  system,  and  during  this 
period  the  system  is  gaining  kinetic  energy ;  when  the  electro- 
motive force  ceases  this  kinetic  energy  is  employed  in  carrying 
the  current  on  against  the  resistance  of  the  circuit ;  thus  we 
may  look  upon  electricity  in  motion  as  having  kinetic  energy. 

\/  Again,  a  charged  body  possesses  potential  energy  measured 
as  we  have  seen  by  half  the  product  of  its  charge  and  its 
potential. 

Thus  any  charged  electrical  system  resembles  a  dynamical 
system  in  being  capable  of  having  both  potential  and  kinetic 
energy.  If  for  example  the  system  be  a  charged  condenser, 
and  the  plates  be  connected  by  a  wire,  the  potential  energy  of 
the  charge  becomes  kinetic  energy  in  the  current  of  discharge, 
and  is  transformed  into  heat  by  the  resistance  of  the  wire. 
We  wish  to  study  these  changes  a  little  more  fully. 

Suppose  we  have  a  smooth  ball,  Fig.  261,  resting  in  a 
groove  on  a  horizontal  table,  and  attached  at  opposite  ends  of 
a  diameter  to  two  springs  or  pieces  of  elastic,  which  we  will 
suppose  are  originally  slightly  stretched,  and  have  their  ends 
fixed  to  the  opposite  ends  of  the  groove. 


Fig.  261. 

Displace  the  ball  in  the  groove  so  as  to  stretch  one  of  the 
springs  further,   relaxing  at  the   same  time  the  other.     On 


255]  ELECTRIC   WAVES  413 

releasing  the  ball  it  will,  if  the  groove  be  smooth,  move  back 
to  and  past  its  equilibrium  position,  stretching  the  spring 
which  was  slack  and  relaxing  the  other ;  it  will  then  come 
to  rest  for  a  moment,  and  then  reversing  its  motion  move 
back  through  the  equilibrium  position  towards  the  point  from 
which  it  started.  A  series  of  oscillations  is  thus  set  up  which 
would  go  on  continuously  if  the  system  were  quite  free  from 
frictional  resistance,  but  which  in  reality  will  die  down  at  a 
rate  depending  on  the  friction.  The  ball  before  being  released 
has  potential  energy ;  this  is  transformed  into  kinetic,  and 
again  back  into  potential. 

Consider  a  ball  hanging  from  the  end  of  a  fine  vertical 
thread,  Fig.  262.  Raise  the  ball  slightly,  still  keeping  the 
thread  stretched;  in  its  raised  posi- 
tion it  has  potential  energy.  Then 
release  the  ball;  it  swings  down, 
gaining  kinetic  energy  but  losing 
potential,  through  its  position  of 
equilibrium,  where  for  a  moment 
it  has  its  maximum  kinetic  energy, 
rising  on  the  other  side,  gaining 
potential  and  losing  kinetic  energy. 
This  goes  on  for  some  time  in  the 
open  air,  and  would  continue  for 
ever  if  we  could  entirely  remove 
the  frictional  resistance  caused  by  Fig.  262. 

the  air.     If  now  we  try  to  repeat 

the  same  experiment  in  water,  the  number  of  oscillations  of  the 
ball  before  it  comes  to  rest  is  greatly  reduced,  while  probably 
in  oil  or  some  other  viscous  medium  it  will  not  oscillate 
at  all  but  sink  slowly  to  its  equilibrium  position. 

A  charged  condenser  corresponds  to  the  raised  ball  in 
having  potential  energy.  If  we  connect  the  condenser  plates 
by  a  wire  we  allow  a  current  to  flow,  the  energy  takes  the 
kinetic  form,  and  just  as  with  the  ball,  if  the  frictional 
resistance  be  small,  we  obtain  oscillations,  so  by  sufficiently 
reducing  the  resistance  of  the  electrical  system  we  obtain 
oscillations  of  electricity.  The  positive  charge  of  the  con- 
denser passes  through  the  wire  from  one  plate  to  the  other, 
the  plate  which  was  positive  becoming  negative,  and  vice  versd, 


414  ELECTRICITY  [CH.  XXVI 

and  this  may  continue  for  some  time ;  the  current  in  the 
wire  starts  from  zero,  rises  to  a  maximum,  and  then  dies 
down  to  zero,  after  which  it  starts  again  in  the  opposite 
direction ;  and  these  alternations  may  go  on  for  some  time. 

If  however  the  resistance  of  the  wire  be  very  large  the 
number  of  oscillations  will  be  very  small,  in  fact  there  may 
be  none,  the  condenser  is  discharged  at  once  without  any 
alternations  of  sign  in  its  plates. 

256.  Transmission  of  Magnetic  Force.     Now  in 

the  neighbourhood  of  any  conductor  carrying  a  current  there 
is  a  magnetic  field  which  is  proportional  to  the  current ;  and 
when  the  current  is  an  alternating  one  the  field  at  each  point 
is  alternating  also. 

Consider  then  a  point  at  a  distance  from  the  conductor ; 
we  may  ask  ourselves  the  question  how  are  changes  in  the 
magnetic  field  related  to  those  of  the  current ;  do  they  occur 
simultaneously,  or  is  there  a  definite  interval  between  the 
moment  at  which  a  change  takes  place  in  the  current  and 
that  at  which  the  corresponding  change  occurs  in  the  electric 
or  magnetic  field  at  the  point  P'l 

It  has  been  shewn  that  there  is  a  definite  interval  of  time 
between  these  two  moments,  and  further  that  this  interval  is 
proportional  to  the  distance  between  the  current  and  P.  The 
effect  moves  with  a  uniform  speed. 

The  changes  in  the  electric  and  magnetic  force  are  pro- 
pagated outward  from  the  alternating  current  and  travel 
through  space  with  a  definite  velocity. 

257.  Electric    Waves.       Moreover    experiment    has 
shewn  that  this  velocity  of  the   electric  waves  is  the  same 
as  that  of  light,  about  300,000  kilometres  or  186,000  miles 
per  second. 

The  period  of  the  waves,  the  rapidity  with  which  they 
succeed  each  other,  will  be  the  same  as  the  period  of  the 
alternations  of  the  current,  and  will  depend  on  the  form 
and  dimensions  of  the  electric  circuit.  Lord  Kelvin  and 
von  Helmholtz  both  shewed  that  if  L  measures  the  coefficient 


255-257] 


ELECTRIC   WAVES 


415 


of  self-induction  of  the  circuit,  and  C  the  capacity  of  the 
condenser,  then  the  period  of  oscillation,  provided  the  re- 
sistance is  very  small,  is  equal  to  the  value  2?r\/ LC. 

The  theory  that  changes  in  electric  and  magnetic  force 
are  propagated  through  space  with  a  velocity  equal  to  that 
of  light  is  due  to  Clerk  Maxwell,  while  Hertz  was  the  first 
to  verify  the  theory  by  direct  experiment. 

To  do  this  Hertz  had  first  to  produce  oscillations,  and 
then  to  observe  the  electric  or  magnetic  changes  which  take 
place  at  a  distance.  To  excite  the  electric  waves  Hertz  used 
two  square  plates,  Fig.  263,  each  40  cm.  in  edge,  connected 


Fig.  263. 

by  wires  about  30  cm.  long  to  two  small  spherical  knobs. 
These  were  gilt  and  placed  some  2  or  3  cm.  apart.  The 
plates  constitute  the  condenser,  the  wire  the  circuit  in  which 
the  current  flows.  The  condenser  is  charged  by  being 
connected  to  an  induction  coil  or  an  electrical  machine. 
When  its  potential  has  risen  sufficiently  the  insulation 
between  the  knobs  breaks  down,  and  an  alternating  discharge 
takes  place ;  in  Hertz's  arrangements  the  period  of  the 
alternation  was  about  1*851  x  10~8  seconds,  that  is  to  say,  if 
the  alternations  continued  regularly  there  would  be  about 
50  millions  of  them  in  a  second ;  in  reality  only  a  few  of 
the  alternations  pass,  they  are  rapidly  damped  out,  the  spark 
across  the  knobs  lasts  for  a  very  short  time,  and  then  the 
plates  are  again  charged  up  by  the  machine. 

To  detect  the  changes  in  the  electric  force  at  a  distance 
Hertz  made  use  of  the  principle  of  resonance. 

If  we  consider  any  mechanical  system  which  can  vibrate 
in  a  definite  period,  then  if  a  series  of  small  impulses  having 
that  same  period  be  applied  to  it,  the  effect  is  much  greater 
than  that  due  to  a  series  of  much  larger  impulses  which  have 
some  different  period. 


416 


ELECTRICITY 


[CH.  XXVI 


Hertz's  receiver,  Fig.  '264,  took  the  form  of  a  wire  circle 
35  centimetres  in  radius,  having  on  its  ends  two  small  knobs 
brought  very  close  together;  the  period  of 
electrical  oscillations  in  such  a  wire  can  be 
shewn  to  be  the  same  as  that  in  the  smaller 
circuit.  Very  small  electrical  impulses  of 
this  period  applied  to  the  wire  set  up  oscil- 
lations, and  sparks  which  can  be  observed 
pass  across  the  gap. 

Thus  Hertz  was  able  to  set  up  electrical 
waves  which  travelled  out  from  his  emitter  with  the  velocity 
of  light,  and  to  detect  them  at  a  distance  with  his  receiver. 


Fig.  264. 


258.  Wireless  Telegraphy.  These  experiments  form 
the  basis  of  wireless  telegraphy. 

Instead  of  Hertz's  ring,  some  form  of  coherer  is  usually 
employed  as  a  receiver. 

It  was  shewn  first  by  Branly  that  the  electric  resistance 
of  a  glass  tube  tilled  with  iron  filings  is  very  much  reduced 
if  electric  oscillations  fall  on  it.  If  the  tube  be  tapped  or 
slightly  shaken  it  recovers  its  greater  resistance. 

When  used  as  a  receiver  by  Lodge  it  was  connected  up  in 
series  with  a  cell  and  a  galvanometer,  Fig.  265 ;  in  its  normal 


Fig.  265. 

condition  the  coherer  resistance  is  so  large  that  the  current 
passing  is  very  small.  When  electric  waves  fall  on  it  the 
resistance  falls,  and  the  galvanometer  needle  is  deflected  ;  the 
sensitive  condition  is  restored  by  tapping  the  board  on  which 
the  coherer  rests. 


•  B 

-•E 


257-258]  ELECTRIC   WAVES  417 

Wireless  telegraphy  as  now  practised  is  a  consequence  of 
the  results  of  these  arid  similar  experi- 
ments. An  oscillator  usually  of  the  form 
devised  by  Righi  is  employed.  In  Fig. 
266  A  and  B  are  two  small  spheres  or 
cylinders  placed  close  together  on  an 
insulating  support.  D  and  E  are  two 
other  spheres  placed  near  them,  and 
connected  to  the  secondary  terminals  of 
an  induction  coil  or  electrical  machine  C.  j^  266. 

On  working  the  machine  a  spark  passes 

from  D  to  E  through  the  spheres  A  and  B  and  oscillations 
are  set  up  which  depend  on  these  spheres. 

In  order  to  concentrate  the  waves  at  the  receiving  station 
on  to  the  coherer  or  other  receiver  a  tall  pole  carrying  a  wire  is 
employed.  The  current  through  the  coherer  works  a  relay 
which  drives  a  Morse  instrument  and  thus  signals  can  be 
received  and  read ;  an  electromagnetic  arrangement  is 
employed  to  tap  the  coherer  at  frequent  intervals  and  thus 
keep  it  in  the  sensitive  state. 

In  his  most  recent  work  M.  Marconi  has  employed  a 
different  form  of  receiver. 

The  energy  needed  to  transmit  electric  waves  to  great 
distances  is  very  large,  hence  both  the  transmitting  and 
receiving  apparatus  are  of  necessity  on  a  very  great  scale. 


G.  E. 


CHAPTER  XXVII. 

TRANSFERENCE  OF  ELECTRICITY  THROUGH  GASES  ; 
CORPUSCLES  AND  ELECTRONS. 

259.     Electric   Discharge  through   Gases.      The 

passage  of  electricity  through  highly  rarefied  gases  has  afforded 
a  field  for  a  very  large  amount  of  investigation.  The  gas  is 
usually  contained  in  a  glass  tube,  Figure  267,  into  the  two  ends 


Fig.  267. 

of  which  platinum  wires  are  sealed.  These  can  be  connected  to 
the  secondary  terminals  of  an  induction  coil,  or  to  a  battery 
of  a  large  number  of  cells,  and  the  consequences  of  varying 
the  nature  or  the  pressure  of  the  gas  in  the  tube  examined. 

It  is  convenient  for  some  purposes  to  connect  the  ter- 
minals also  to  two  insulated  metal  balls  outside  the  tube  ;  by 
varying  the  distance  between  these  balls  the  spark  may  be 
made  to  pass  at  will,  either  through  the  tube,  or  through  the 
air  between  the  balls,  and  the  distance  between  the  balls  at 
which  it  just  ceases  to  pass  through  the  air  and  begins  to  pass 
through  the  tube  affords  a  measure  of  the  resistance  offered  by 
the  gas  in  the  tube  to  the  passage  of  the  spark. 


259-260]  TRANSFERENCE   OF   ELECTRICITY  419 

For  many  purposes  it  is  best  to  employ  as  the  negative 
electrode  or  kathode  a  small  flat  plate  of  aluminium.  The 
tube  is  connected  to  a  mercury  pump  in  such  a  way  that  the 
gas  it  contains  can  be  reduced  to  a  high  degree  of  rarefaction. 

At  first  when  the  pressure  is  high  the  current  does  not 
pass  through  the  tube,  as  the  pressure  is  reduced  it  begins  to 
pass,  and  a  narrow  luminous  line  is  seen  down  the  centre  of 
the  tube  ;  the  particles  of  gas  along  the  axis  are  intensely 
heated  by  the  passage  of  the  current  and  become  luminous. 
On  reducing  the  pressure  further  the  glow  appears  to  fill  the 
whole  tube.  When  the  pressure  is  still  further  reduced  to 
that  due  to  from  J  to  |  of  a  millimetre  of  mercury  further 
changes  take  place.  The  luminous  column  is  broken  up  near 
the  positive  electrode  into  a  series  of  portions  separated  by 
dark  strise  ;  immediately  round  the  negative  electrode  is  a 
soft  luminous  glow,  and  between  it  and  the  striated  column 
a  comparatively  dark  space  known  as  Faraday's  dark  space  ; 
careful  observation  shews  that  this  negative  glow  is  separated 
from  the  kathode  by  a  second  dark  space  at  present  very 
narrow. 

Carry  the  exhaustion  further,  the  column  of  striae  con- 
tracts and  is  followed  by  Faraday's  dark  space,  the  negative 
glow  extends  out  into  the  tube,  and  the  dark  line  between  it 
and  the  kathode  widens,  becoming  what  is  known  as  Orookes' 
dark  space.  On  carrying  the  exhaustion  still  further  Crookes' 
dark  space  fills  the  tube,  the  negative  glow  is  confined  to 
a  small  space  near  the  anode ;  the  centre  of  the  tube  is 
dark  but  the  surface  of  the  tube  glows  with  a  phosphorescent 
light,  the  colour  of  which  depends  on  the  nature  of  the  glass, 
with  lead  glass  it  is  blue,  with  soda  glass  it  is  green. 

26O.  Crookes'  Tubes.  The  phenomena  which  occur 
in  the  tube  in  this  condition  were  first  investigated  by 
Crookes. 

He  was  led  to  the  belief  that  in  the  highly  rarefied 
condition  within  the  tube  particles  of  matter  charged  with 
negative  electricity  were  shot  off  from  the  neighbourhood  of 
the  kathode  at  a  great  velocity  and  travelled  in  straight  lines 


through  the  tube. 


27—2 


420  ELECTRICITY  [CH.  XXVII 

Thus  if  an  object  such  as  a  cross  cut  out  of  any  material 
be  placed  near  one  end  of  a  tube,  as  in  Fig.  268,  usually  pear- 
shaped  as  in  the  figure,  with  the  kathode  at  the  narrow  end 


Fig.  268. 

a  shadow  of  the  cross  is  projected  on  to  the  thick  end  of  the 
tube.  Again  Crookes  mounted  in  the  tube  a  small  wheel  with 
vanes,  arranging  it  so  that  the  stream  from  the  kathode — the 
kathode  rays — might  strike  the  vanes  on  one  side  of  the  axis 
only  when  the  wheel  was  set  in  rapid  rotation. 

If  the  kathode  is  made  concave  so  as  to  cause  the  rays 
which  leave  it  at  each  point  in  the  direction  of  the  normal  at 
that  point  to  converge,  then  a  small  body  placed  at  the  focus 
or  point  of  convergence  is  greatly  heated;  the  temperature 
can  by  this  means  be  raised  sufficiently  high  to  melt  platinum. 

Various  substances,  such  as  some  of  the  rare  earths,  placed 
within  the  tube  glow  with  most  brilliant  phosphorescence 
where  the  rays  strike  them. 

261.  Kathode  Rays.  The  kathode  rays  constitute 
a  current  of  negative  electricity  streaming  from  the  negative 
electrode,  and  the  above  phenomena  shew  that  there  is  a  stream 
of  matter  which  must  be  in  a  very  finely  divided  state. 

The  kathode  rays  can  be  deflected  by  a  magnet.  This  is 
obvious  at  once  :  on  bringing  a  magnet  near  the  stream  it 
is  deflected,  just  as  it  should  be  according  to  known  laws  if 
it  were  a  current  flowing  in  a  perfectly  flexible  conductor. 


260-262]  TRANSFERENCE   OF   ELECTRICITY  421 

A  screen  with  a  narrow  horizontal  slit  is  placed  in  front  of 
the  kathode  ;  the  rays  passing  through  the  slit  form  a  narrow 
horizontal  beam  and  produce  a  patch  of  light  on  the  wall  of 
the  tube  opposite  to  the  kathode.  As  the  magnet  approaches 
the  patch  of  light  moves.  If  the  kathode  is  to  the  right  as 
shewn  in  Fig.  269,  and  the  lines  of  magnetic  force  pass  through 
the  paper  from  above  downwards,  the  patch  of  light  is  raised, 
if  the  direction  of  the  lines  of  magnetic  force  be  reversed  it  is 
lowered. 


QJD.9J 


Fig.  269. 

The  fact  that  the  kathode  rays  were  carriers  of  negative 
electricity  was  established  by  J.  J.  Thomson. 

He  placed  inside  a  Crookes'  tube  two  metallic  tubes,  one 
within  the  other ;  the  outer  one  was  connected  to  earth,  the 
inner  one  was  insulated  and  connected  to  an  electrometer. 
Two  narrow  slits  were  made  in  the  walls  of  the  tubes  opposite 
to  each  other,  and  by  means  of  a  magnet  the  kathode  stream 
which  when  undeflected  passed  outside  both  tubes  could  be 
directed  into  the  inner  tube.  Here  being  inside  a  hollow 
closed  conductor  the  stream  gave  up  its  electricity  to  the  con- 
ductor, and  the  amount  of  charge  acquired  by  the  tube  was 
measured  by  the  electrometer.  The  tube  always  acquired  a 
negative  charge  ;  the  outer  earthed  tube  served  to  keep  off 
stray  electrification. 

If  we  suppose  that  this  negative  electrification  is  carried 
by  particles  and  call  n  the  number  of  the  particles  which 
enter  the  tube  in  a  given  time,  e  the  charge,  and  m  the  mass 
of  each,  then  an  experiment  such  as  the  above  enables  us  to 
find  the  value  of  ne. 

262.  Ionic  Charge  in  Electrolysis.  When  treat- 
ing of  electrolysis  we  were  led  to  consider  definite  quantities 


422  ELECTRICITY  [CH.  XXVII 

of  matter  each  carrying  a  definite  charge  ;  to  these  we  gave  the 
name  of  ions,  and  we  saw  that  the  ionic  charge,  the  quantity 
of  electricity  carried  by  any  ion,  was  a  constant.  Moreover 
we  have  learnt  that  the  mass  of  hydrogen  liberated  by  the 
passage  of  an  electromagnetic  unit  quantity  of  electricity  is 
l'03xlO~4  gramme.  Calling  this  10~4  gramme,  since  the 
ratio  of  the  mass  liberated  to  the  electricity  set  free  is  a 
constant  for  the  same  substance,  we  have  if  e  be  the  charge 
and  m  the  mass  of  a  single  hydrogen  ion  m/e  equal  to  10~4. 


Hence  for  hydrogen  ions 


-  =  104. 
m 


If  we  know  the  mass  of  an  ion  of  hydrogen  we  can  calculate 
from  this  its  charge. 


263.  Charge  carried  in  Kathode  Rays.  Again,  if 
we  assume  e  to  be  the  charge  and  m  the  mass  of  the  carriers  of 
negative  electricity  in  the  kathode  rays,  a  relation  between 
these  quantities  and  u  the  velocity  with  which  they  move  can 
be  obtained  by  observing  the  curved  path  of  the  stream  in  the 
magnetic  field.  When  the  field  is  at  right  angles  to  the  kathode 
stream  that  path  is  seen  to  be  a  circle,  and  its  radius  r  can 
be  measured. 

Let  u  be  the  velocity  of  any  particle,  then  since  mi  is  its 
mass,  the  force  which  must  be  applied  to  make  it  describe  a 
circle  of  radius  r  is  mu*/r.  But  this  force  arises  from  the 
magnetic  field  and  is  equal  to  its  strength  multiplied  by  the 
strength  of  the  current.  The  current  strength,  considering  a 
stream  1  square  centimetre  in  section,  is  eu,  and  if  H  be  the 
magnetic  intensity  the  force  tending  to  make  the  particles 
describe  the  circular  paths  is  Heu. 

mu2      T. 

Hence  —  =  lieu 

r 

m 
or  —  u  =  Hr. 

e 

Now  we  do  not  know  the  value  of  u  but  experiment  proved 


262-263]  TRANSFERENCE    OF    ELECTRICITY  423 

Hr  to  have  the  value  3  x  106  approximately.     If  we  assume  u 
to  be  the  velocity  of  light  or  3  x  1010cm.  per  second,  we  obtain 


=  10-', 

e 

the  same  value  as  previously. 

Thus  the  facts  were  consistent  with  the  assumption  that 
in  the  kathode  rays,  particles  having  the  mass  of  the  ions  known 
to  us  in  electrolysis  were  shot  from  the  negative  electrode 
with  the  velocity  of  light. 

On  the  other  hand  many  of  the  phenomena  of  the  dis- 
charge in  a  Crookes'  tube  did  not  depend  on  the  nature  of  the 
gas,  as  they  should  have  done  if  the  moving  particles  were  ions, 
and  it  was  difficult  to  imagine  particles  as  massive  as  the  ions 
moving  with  the  velocity  of  light. 

Hence  J.  J.  Thomson  was  led  to  enquire  further. 

We  have  seen  how  he  measured  the  total  charge  given  up 
per  second  by  the  particles.  Let  this  be  Q  and  let  the  number 
of  particles  per  cubic  centimetre  of  the  stream,  assumed  to  be 
of  unit  cross  section,  be  N.  Then  the  number  entering  the 
inner  tube  per  second  is  JVu,  and  their  charge  is  Nue. 

Hence  we  have 

Neu  =  Q,  and  Q  is  known. 

Again,  these  particles  give  up  their  energy  to  the  inner  tube  ; 
by  treating  the  tube  as  a  calorimeter  of  known  heat  capacity 
and  measuring  the  rise  of  temperature  per  second  due  to  the 
bombardment,  we  can  find  W,  the  energy  given  up  by  the 
particles.  But  this  energy  is  ^N'mn3,  for  the  mass  of  the 
entering  particles  is  Nmu  arid  the  velocity  of  each  is  u. 

Thus  we  have  the  three  equations, 
Neu^Q 
=  W 

mn-Ur. 

e 


424  ELECTRICITY  [CH.  XXVII 

From  the  first  two  equations  we  obtain 
mu*      2  W 


AV) 

Combining  this  with  the  equation  —  u  =  Hr  we  find 
2  IF 


On  substituting  the  experimental  values  on  the  right 
hand  it  appears  that  u  was  about  equal  to  109  while  the 
value  for  m/e  came  to  be  about  10~7  instead  of  10~4  as 
obtained  from  electrolysis. 

Thus  we  are  bound  to  assume  either  that  m  is  much 
less,  or  e  much  greater  than  the  corresponding  electrolytic 
quantities — of  course  both  facts  may  be  true.  If  however  we 
assume  e  to  be  the  same,  then  m  is  about  1000th  part  of  the 
mass  of  an  ion  of  hydrogen. 

264.    Number  of  Particles  in  the  Kathode  Rays. 

The  value  thus  obtained  for  the  ratio  m/e  was  substantiated 
by  the  result  of  a  number  of  other  experiments.  Before, 
however,  it  is  possible  to  determine  which  of  the  above 
hypotheses  as  to  the  values  of  m  and  e  separately  is  the  true 
one,  it  is  necessary  to  determine  N  the  number  of  particles 
per  unit  volume. 

This  also  has  been  done  by  Thomson.  Aitken  had  shewn 
that  a  cloud  is  ordinarily  formed  by  the  condensation  of  water 
vapour  about  small  nuclei  of  dust  or  other  foreign  material  in 
the  atmosphere,  and  C.  T.  R.  Wilson  has  proved  that  the 
particles  which  serve  as  the  carriers  of  negative  electricity  in 
the  kathode  rays  would  serve  also  as  such  nuclei. 

If  now  a  cloud  be  formed,  by  allowing  the  moist  air  in  a 
closed  vessel  to  expand  suddenly,  and  then  permitted  to  settle, 
the  nuclei  are  cleared  away  by  the  cloud,  and  so  long  as  no 
fresh  nuclei  are  formed,  no  further  clouds  will  be  produced 
in  the  space. 

Thus  the  number  of  nuclei  in  the  space  will  be  the  same 
as  the  number  of  drops  in  the  cloud. 

The  total  volume  of  the  drops  formed  can  be  calculated 


263-267]  TRANSFERENCE   OF   ELECTRICITY  425 

from  a  knowledge  of  the  expansion  producing  the  cloud ; 
hence  if  we  can  ^calculate  in  any  way  the  size  of  the  drops 
we  can  obtain  their  number  by  dividing  the  total  volume  by 
the  volume  of  each  drop. 

The  drops  settle  down  at  a  uniform  rate  because  the 
resistance  due  to  the  air  just  balances  the  weight  of  the  drop  ; 
and  a  relation  can  be  found  between  the  velocity  of  a  drop 
which  has  attained  this  terminal  speed  and  its  radius.  From 
this  relation  it  is  possible  to  calculate  the  radius  of  a  drop, 
and  hence  the  number  of  drops  in  each  cubic  centimetre. 

The  result  of  J.  J.  Thomson's  experiments  was  to  shew 
that  the  number  of  drops  per  c.c.  under  the  conditions  of  his 
experiment  was  4  x  104,  and  that  under  the  same  circumstances 
the  value  of  e,  the  charge  carried  by  each  particle,  is  6'3  x  10~10 
electrostatic  units. 

265.  Charge  of  an  Ion  in  Electrolysis.     Now  we 

have  seen  that  the  charge  carried  by  one  gramme  of  hydrogen 
in  electrolysis  is  about  104  electromagnetic  units  of  electricity; 
more  accurately  it  is  9650  units  ;  the  number  of  hydrogen 
ions  in  a  gramme  can  be  determined  by  means  of  experiments 
on  the  viscosity  of  hydrogen,  and  is  found  to  he  4'4  x  1  <)-'•",  and 
the  number  of  electrostatic  units  in  one  electromagnetic  unit 
is  3  x  1010.  Thus  the  charge  on  a  hydrogen  ion  in  electrostatic 
units  is  9650  x  3  x  1010/44  x  1020  or  6-5  x  10~'°  electrostatic 
units,  the  same  practically  as  that  carried  by  the  particles 
in  the  kathode  discharge. 

266.  Corpuscles.     Again  we  have  seen  that  for  hydro- 
gen ions  the  ratio  m/e  is  about  10~\  while  for  the  kathode 
particles  it  is   10~7  ;    more   exactly    the    value  is    1-4  x  10~7. 
Hence  the  mass  of  the  kathode  ray  particle  is  about  1'4  x  10~3 
(or  1 /700th)  of  that  of  an  hydrogen  ion. 

J.  J.  Thomson  calls  these  particles  corpuscles ;  other 
writers  have  given  them  the  name  of  electrons,  and  the  most 
modern  theory  of  electricity  is  based  on  the  supposition  that 
practically  all  the  phenomena  of  electricity  are  due  to  the 
presence  and  motion  of  electrons. 

267.  Electrons.      Electrons    can    he   freed    by   other 
means  than  the  kathode  rays. 


426  ELECTRICITY  [CH.  XXVII 

Thus  Lenard  shewed  that  if  the  kathode  rays  be  allowed 
to  fall  on  a  window  of  thin  aluminium  sealed  into  the  walls  of 
a  vacuum  tube,  rays  having  much  the  same  properties  pene- 
trate into  the  air  outside  the  tube ;  we  may  suppose  either 
that  some  of  the  electrons  penetrate  through  the  aluminium 
or  that  the  impact  of  the  electrons  inside  the  tube  is  sufficient 
to  knock  off  electrons  from  the  outer  side  into  the  air. 

268.  Rontgen  Rays.  Under  the  same  circumstances 
as  these,  Rontgen  had  shewed  that  another  system  of  rays  are 
produced  in  the  air. 

The  Rontgen  or  X  rays  pass  through  glass  and  many  other 
materials  very  easily.  When  they  fall  on  certain  fluorescent 
substances  they  excite  these  strongly,  they  can  penetrate 
readily  substances  like  paper,  wood,  and  light  materials  which 
are  opaque  to  light.  The  denser  metals  are  more  opaque ; 
their  use  in  surgery,  owing  to  the  fact  that  they  penetrate  the 
muscle  and  lighter  tissues  of  the  body  while  the  bones  are 
opaque  to  them,  is  well  known.  They  are  best  produced  by 
the  use  of  a  tube  of  soda  glass  of  the  shape  shewn  in  Fig.  270. 


Fig.  270. 

The  kathode  is  of  aluminium  and  is  concave.  The  kathode 
rays  are  shot  on  to  the  platinum  anode  inclined  at  45°  to  the 
direction  in  which  they  are  travelling  and  from  this  the 
Rontgen  rays  appear. 

According  to  the  theory  of  Stokes,  developed  by  Thomson, 
each  electron  as  it  reaches  the  anode  sets  up  a  sudden  impulse 
in  the  ether  which  travels  outwards  as  a  single  wave  from  the 
anode ;  these  impulses  follow  each  other  rapidly  but  not  with 
the  rapidity  or  regularity  requisite  to  produce  a  continuous 
stream  of  light  waves ;  the  properties  of  such  a  discontinuous 
series  of  impulses  will  resemble,  as  Thomson  proved,  those  of 
the  Rontgen  rays. 


267-270]  TRANSFERENCE   OF    ELECTRICITY  427 

269.  Production  of  Electrons.  Electrons  are  produced 
by    the    action    of    Rontgen    rays;    thus  air    through    which 
Rontgen  rays  are  allowed  to  pass  loses  its  insulating  power 
and    becomes    conducting,   owing    to  the  production   of   free 
electrons,  each  carrying  its  negative   charge.     They  are  also 
produced    by  the   action  of  ultra-violet   light   011  negatively 
charged  bodies.     A  body  when  charged  negatively,  loses  its 
charge  rapidly  when  illuminated  by  ultra-violet  light.      Some 
substances — as  salts  of  Uranium,  Radium  and  certain  salts  of 
Thorium — give  off  radiation  continuously  and  a  part  of  the 
radiation  which   they   emit   has   the  properties  of   electrons. 
These  radiations  have  been  studied  by  M.  H.  Becquerel,  their 
discoverer,    M.   and   Madame   Curie,    Professors    Rutherford, 
Townsend  and  others. 

It  is  to  be  noticed  that  in  all  cases  the  electrons  or 
corpuscles  are  associated  with  a  negative  charge  of  electricity ; 
streams  of  positively  charged  matter  may  exist,  but  the  masses 
concerned  are  comparable  with  those  of  the  atoms,  instead  of 
being  nearly  one  thousand  times  smaller  than  hydrogen  atoms. 

270.  Electron  Theory  of  Electricity.     According 
to  the  electron  theory  a  neutral  atom  consists  of  an  electron 
or   series    of    electrons    each    carrying    its    negative    charge 
together  with  a  positively  charged  nucleus,  the  total  positive 
charge  being  equal  to  the  sum  of  the  negative  charges  on  the 
electrons. 

It  is  possible  in  various  ways  to  attach  one  or  more 
electrons  to  such  an  atom  ;  it  then  becomes  negatively  charged ; 
it  is  also  by  hypothesis  possible  to  detach  one  or  more 
electrons;  the  remainder — the  coelectron  as  Prof.  Fleming1 
has  called  it — remains  positively  electrified. 

A  univalent  atom,  like  hydrogen,  is  one  which  can  receive 
or  give  up  one  electron  and  no  more.  A  divalent  atom  can 
receive  or  give  up  two  electrons  and  so  on. 

1  Popular  Science  Monthly,  May,  1902.  Prof.  Fleming's  article  may  be 
referred  to  by  those  who  wish  to  learn  more  of  the  Theory,  see  also  a 
lecture  by  Sir  0.  J.  Lodge,  delivered  before  the  Institution  of  Electrical 
Engineers,  November  27,  1002. 


428  ELECTRICITY  [CH.  XXVII 


A  current  of  electricity  is  a  stream  of  electrons,  a 
through  which  the  electrons  pass  freely  is  a  conductor  ;  within 
a  non-conductor  they  cannot  move  about  readily.  A  gas  may 
be  non-conducting,  because  of  the  absence  of  electrons,  if  they 
are  introduced  it  gains  conductivity.  All  the  phenomena  of 
electric  discharge  and  current  are  convection  phenomena. 

When  electromotive  force  is  applied  to  a  conductor,  the 
electrons  are  urged  through  the  conductor  ;  if  it  be  a  gas  at 
low  pressure  they  stream  from  the  kathode  as  the  kathode 
rays. 

In  an  electrolyte  in  solution,  some  of  the  free  ions  are 
positive  ;  they  are  coelectrons  and  the  electrons  which  have  left 
them  have  joined  on  to  other  ions,  making  them  negative  ; 
there  is  probably  a  continual  interchange  going  on,  but  on  the 
average  the  above  statement  represents  the  position. 

The  negative  ions  are  driven  by  the  E.M.F.  to  the  anode, 
the  positive  ions  travel  to  the  kathode. 

In  a  solid  conductor  the  same  kind  of  separation  and 
combination  of  ions  and  electrons  is  taking  place  but  the  ions 
are  not  free  to  move  ;  the  current  is  conveyed  by  the  electrons 
moving  on  from  ion  to  ion  through  the  solid  ;  the  solid  is 
porous  to  them  but  not  to  the  ions. 

271.  Electrons  and  Galvanic  Action.  There  is 
in  general  a  tendency  for  electrons  to  be  set  free  at  the 
common  surface  of  any  two  bodies.  Thus  with  zinc  in  oxygen 
an  electric  double  layer  is  formed,  the  electrons  in  the  layer  of 
oxygen  round  the  zinc  are  turned  towards  the  zinc  and  are 
opposed  by  coelectrons  in  the  outer  layer  of  zinc  ;  the  same  is 
true  with  copper,  but  the  attraction  of  the  zinc  for  the  oxygen 
electrons  is  greater  than  that  of  the  copper  ;  we  obtain  the 
theory  of  the  voltaic  cell  already  developed. 

In  the  case  of  two  insulators,  such  as  glass  and  silk,  the 
conditions  are  the  same,  over  each  in  air  or  oxygen  there 
is  a  layer  of  positive  electricity  faced  by  the  electrons  of  the 
oxygen. 

When  the  glass  and  silk  are  rubbed  together  the  action 
mixes  up  these  two  double  layers,  some  electrons  are  dragged 


270-273]  TRANSFERENCE   OF    ELECTRICITY  429 

oft'  the  glass  and  retained  by  the  silk,  the  one  becomes  positive, 
the  other  negative. 

272.  Electrons    and    Magnetism.       An    electric 
current  is  as  we  have  said  a  convection  stream  of  electrons. 
J.    J.    Thomson    has    proved    mathematically    that    a    single 
electron  in  motion  produces  magnetic  force,  the  lines  of  force 
are  at  each  moment  circles  whose  centres  lie  in  the  direction 
of  motion  of  the  particle  at  that  moment  and  whose  planes 
are  perpendicular  to  that  direction. 

From  this  follow  the  laws  of  electromagnetic  action,  while 
permanent  magnets  are  whirls  of  electrons. 

273.  Electron  Theory  of  Matter.     We  at  present 
have  but  little  idea  as  to  what  an  electron  is ;  we  may  look 
upon  its  charge  as  a  natural  unit  of  electricity  and  account  for 
most  of  the  phenomena  of  electricity  with  marked  success  by 
considering  the  results  which  follow  from  the  motions  of  such 
a  charged  corpuscle.     Some  have  gone  so  far  as  to  express  the 
belief  that  an  electron  is  the  centre  of  a  particular  kind  of 
motion  in  the  ether,  and  that  matter  is  made  up  of  assemblages 
of  electrons,  but  the  discussion  of  such  questions  would  take 
us  very  wide  of  our  limits  which,  indeed,  have  been  consider- 
ably exceeded  already. 


ANSWERS   TO  EXAMPLES. 

ELECTROSTATICS. 

CHAPTER  VI.     PAGE  101. 

15.     3,000   electrostatic  units.  16.     90  electrostatic  units. 

17.  (a)   Force  of  attraction  equal  to  2  dynes. 
(b)    Force  of  repulsion  equal  to  -25  dyne. 

18.  (1)   The  sphere  c  should  be  placed  in  the  line  of  the  spheres  a 

and  b;  on  the  opposite  side  from  a  and  141-4  cms.  from  b. 
(2)   In  the  line  of  the  spheres;   same  side  as  a  and  141 '4  cms. 
from  b. 

19.  (a)    Zero  position  lies  between  the  spheres  at  a  distance  of  f  metre 

from  more  highly  charged  sphere. 

(b)    Zero  position  in  line  of  spheres ;  200  cms.  from  more  highly 
charged  sphere,  and  100  cms.  from  the  lesser  charged  one. 

20.  24/y/ry  electrostatic  units.     //  =  gravitational  force  in  dynes. 

23.     (1)    The  same  Q  units.  (2)   Potential  =  Q  (-  +  -  +  - }. 

\Cl        c-2        C3/ 

25.  Charges  :  Large  sphere  6  times  the  charge  of  smaller  one. 
Potentials:  ,,          ,,       same  potential  as             ,,        „ 
Densities:  ,,          ,,       £  the  density  of                ,,         ,, 
Energies:  ,,          .,       6  times  the  energy  of      ,,         ,, 

26.  125  ergs.  27.     200  electrostatic  units. 


29.     Loss  in  energy  of  c  =  %  cv'2    1  -  /--.MO    • 

f    ,     £c'cVJ 

Gam  in  energy  of  c  -  *—         . 
\c-\-c  ) 

Total  loss  due  to  redistribution  of  charge  =$cv2 1 1  -  -— /  J  . 


432  ELECTRICITY 

30.     l:f  31.     (1)    Same-0.       (2)    Half=|. 

32.     Total  charge     =  36  electrostatic  units. 

,,     capacity  =9  ,,  ,, 

Final  potential^  4  „  ,, 

35.  202-5  electrostatic  units. 

36.  («)    1279-95  units  of  capacity. 

(b)   Capacity  increased  k  times,  where  k  =  specific  inductive  capacity 
of  turpentine. 

37.  (1)     Difference  of  potential  —  2  ^/TT  electrostatic  units. 


500 
(2)     Charge  on  either  plate  =  —  ,,  ,, 

VT 

38.  (1)   400  electrostatic  units.     (2)    800  electrostatic  units. 

39.  (1)   837  electrostatic  units.     (2)   418-5  v2  ergs. 

(The  base  of  the  jar  is  assumed  to  be  coated.) 

40.  1  x  10~6  C.G.S.  units  of  heat. 

41.  Difference  of  potential  = 


MAGNETISM. 
CHAPTER  XII.     PAGE  180. 

2.  6  dynes.  3.     Strength  of  pole  =  18  C.G.S.  units. 

4.  Moment  of  couple  =  600.  5.     Moment  of  couple  =  72. 

6.  ^:2>':Casl:2:2.  7.     -0008  dynes. 

8.  14-14  dynes.  10.  '0125  dynes.  H.     1:1-21. 

•  T  ft  71 T 

12.  Forces  of       — dynes  each,  applied  at  the  ends  of  the  magnet; 

(1  =  length  of  magnet);  the  couple  produced  by  the  forces  to 
act  in  opposition  to  the  couple  produced  by  the  Earth's  mag- 
netic force. 

13.  120°  more.  14.     11,250  C.G.S.  units.  15.     1:1-176. 

16.  18-37  oscillations  per  minute. 

17.  >/99'5  =  approx.   10    oscillations    per    minute.      (The  magnet    is 

assumed  to  be  very  long.) 

18.  53-48  C.G.S.  units.  19,     Tangent  of  angle  =  '98. 


ANSWERS   TO    EXAMPLES  433 

VOLTAIC  ELECTRICITY. 
CHAPTER  XIX.     PAGE  316. 

1.  Oxygen  =  8-24xlO~4. 
Iron  (Ferrous)  =28-84  x  10~4. 
Iron  (Ferric)    =  19  '23  x  10~4. 

2.  1-03  xlO-4.  3.     1-1412  gms.  4.     H  gm. 

5.  Copper  =  -7875  gm.     Hydrogen  =  "025  gm.     Oxygen  =  -20  gm. 

6.  (1)    '08  C.G.S.  unit  of  current. 

(2)   8-3  volts  =  8 -3  x  10~8  electromagnetic  units  of  potential. 

7.  Kesistance  of  wire  =  2  ohms. 

,,          ,,  battery  =  1 -2  ohms. 

8.  '4  ohms.  9,     (a)    &  ampere.     (I)    12-5  volts. 

10.     (a)   1-0  volts.     (6)    1-0  ohms.         H.     (a)    1 -5  volts.     (6)    -5  ohms. 

12.  Note  that  Leclanche  cell  has  greater  internal  resistance. 

13.  -75  amps. 

14.  3  rows  in  parallel ;  each  row  containing  4  cells  in  series,  or 
4  „  „  „  „  3 

16.  2  rows  in  parallel,  each  row  containing  5  cells  in  series. 

17.  2  cells  in  parallel,  and  the  third  in  series  with  them. 

18.  TVth  amp. 

19.  (1)    '875  amp.     (2)   52-5  coulombs.     (3)    10-94  volts. 

20.  2  rows  in  parallel,  each  row  containing  24  cells. 

21.  (a)    Current  in  wire  of  1  ohm  resistance  =  -6  amp. 

(b)  „         „         ,,        1'5  ohms       „       =  -4     „ 

(c)  ,,          in  cell  =  1-0  amp. 

22.  (1)     (a)   Current  through  cell  =  £  amp. 

(6)         ,,  ,,         galvanometer  =  I ,  amp. 

(2)     (a)    Current  through  cell  =  %• |  amp. 

(6)         ,,  ,,         galvanometer  =  $$  amp. 

23.  1:2.  24.     Resistance  of  CB  =  70  f  ohms.  25.     revolt. 
26.     -0501  gms  of  Hydrogen.             27.     8^  ohms.              28.     15  ohms. 

29.  12  ohms  ;  7|  ohms;  5*  ohms ;  4f  ohms  ;  1L\  ohms. 

30.  100,000  ohms.  34.     12g  ohms. 
35.  1-3  x  10~9  amps,  for  1  div.  deflexion.          36.     11?,  ohms. 
37.  Specific  Kesistance  =  2  x  104  in  absolute  measure. 

39.     1:4.  40.     l:-747.  41.     17'32  ohms.  42.     1:3. 

43.     -33  amp.          44.     34-9  cms.         45.    —  turns.        46.    286  turns. 
48.     '109  grams  of  copper.  49.     '0182.  51.     18-86  ohms. 

G.  E.  28 


434  ELECTRICITY 

53.  4670  ohms  or  46-7  ohms  ;  the  value  depending  on  the  ratio  coils  to 

which  the  adjustable  resistance  was  adjacent.  If  adjacent  to 
the  10  ohms,  then  resistance  measured  was  4670  ohms;  if  to 
the  100  ohms,  then  resistance  was  46*7  ohms. 

54.  2-5  volts.  55,    40°  C.  56.     2-1  ohms. 

57.  4  rows  in  parallel ;  each  row  containing  2  cells  in  series. 

58.  Current  in  1  ohm  wire=Ji  amps. 

»       ,,  3  „  ,,  =  A  amps. 

Heat  in  1  „  ,,  =  '64  units  per  sec. 

„       „  3  „  „  --21         „         „ 

59.  1:2-49.  60.     13-416  ohms. 

61.     Current  in  thicker  wire  must  be  2-828  times  greater  than  in  thinner 

wire. 
63      -025  Units  of  Heat. 

65.  (a)  5  Horse-power  approx.     (b)    -357°  C.  per  minute. 

66.  1 :  -0025.       67.    (1)    6-5  Kilowatt-Hours.    (2)   -042  Kilowatt-Hours. 
68.     16  sq.  cms. 


ELECTROMAGNETISM   AND   INDUCTION 
OF   CURRENTS. 

CHAPTER  XXIII.     PAGE  387 
1.     -32  C.G.S.  unit. 

ELECTROMAGNETISM  AND   ELECTRO- 
MAGNETIC  INDUCTION. 

CHAPTER  XXIV.     PAGE  404. 

1.  No  effect. 

2.  180°  less   (tan-1  -744  + tan"1  1-524)  =  87°  approx.     Note   that  on 

reversing  the  current,  the  needle  swings  through  180°  owing  to 
the  field  produced  by  the  current  being  greater  than  H. 

3.  6|  dynes.  4.     '0157  dynes. 

*8 
5,     (1)    -r  dynes.   (d  =  perpendicular  distance  in  cms.  betweenthe  wires.) 

(2)    The  force  is  one  of  attraction,  and  acts  at  right  angles  to  both 

wires,  and  in  the  plane  containing  them. 

7.     525  C.G.S.  units.        8.   (1)  7238.     (2)  7238/t.        9.  4-7  x  10~3  volts. 
12.     45-65  C.G.S.  units.     (Assumes  rate  of  change  of  potential  to   be 
uniform.) 


INDEX. 


The  references  are  to  the  pages. 


Absolute  measurement,  247 
Accumulator,  the  Water-dropping, 

85 

Acid-metal  contact,  313 
Action,    Local,    197;    electromag- 
netic, 328,   337 
Action  of  Keepers,  128 
Action  of  a  Magnet  on  a  Current, 

326 

Alphabet,  the  Morse,  408 
Alternating  currents,  measurement 

of,  371 

Ammeters,  364;  soft  iron,  363 
Ampere   Balance,    the,  368;   Lord 

Kelvin's,  369 
Analogy  between  Temperature  and 

Potential,  29 
Annual  Variations,  178 
Arago's  Disc,  382 
Armature,  shuttle  wound,  394 
Armature,  reactions,  399 
Arrangement  of  Batteries  in  series, 

207;  in  multiple  arc  or  parallel, 

208 

Artificial  Magnets,  106 
Astatic  Coil,  331 

Attracted  Disc  Electrometer,  the,  90 
Attraction,  Electric,  3;  explanation 

of,  8,  9 

Attraction,  Magnetic,  108 
Axis  of  a  magnet,  determination  of 

the,  168 

Ballistic  Galvanometer,  the,  279 


Batteries,  Magnetic,  127;  two  fluid, 
202 ;  secondary,  205 ;  in  series, 
207;  in  multiple  arc  or  parallel, 
208;  experiments  on,  255;  in 
series  and  parallel,  262 

Batteries  of  Leyden  Jars,  66 

Battery,  resistance  of  a,  275 ;  Joule's 
Law  applied  to,  290 

Bichromate  Cell,  the,  200 

Boxes,  Resistance,  240 

Bridge,  Wheatstone's,  268 

Bunsen's  Cell,  203 

Calculation  of  Potential,  51 
Calculation  of  Capacity,  63 
Capacity,  Inductive,  45,  62;  of  a 

conductor,  54;    of  a  condenser, 

60;  calculation  of,  63;   unit  of, 

285 
Cell,  Voltaic,  182,  197,  219,   220; 

single  fluid,  200 ;  bichromate,  200 ; 

Leclanche,      201;       dry,      202; 

Grove's,    202;     Bunsen's,    203; 

Daniell's,     203;      Clark's,    204; 

Weston,  205 
Charge  and  Potential,  Relation  of, 

54 
Charged  Conductor,  Potential   of, 

53 ;  energy  of,  55 
Charges  of  Magnetism,  122 
Chemical  Action  of  a  Current,  191 
Chemical  Theories,  312 
Circuit,  magnetic,  347 
Clark  Cell,  the,  204 


436 


INDEX 


Closed  Conductor,  no  electrification 
within,  16 

Closed  Cycle,  Magnetic  Force  due 
to,  134 

Coefficient  of  Mutual  Induction, 
377 

Coil,  Astatic,  331 

Coil  Galvanometer,  361 

Coil,  Induction,  389 

Commutators,  238 

Comparison  of  Kesistances,  264 

Condensers,  58,  281;  explanation 
of  the  action  of,  59 ;  capacity  of, 
60;  energy  of  charged,  62 

Condensing  Electroscope,  68 

Conductance,  213 

Conductor,  no  electrification  within 
closed,  16;  Potential  of  a  charged, 
53;  capacity  of,  54;  heating  by 
a  current,  286 ;  motion  in  a  mag- 
netic field,  337;  in  series,  214; 
in  parallel,  215 

Conductors,  electromagnetic  action 
between,  336 

Conductors  and  Non-Couductors, 
5,6 

Contact,  acid-metal,  313;  potential, 
314 

Contact  experiments,  in  a  vacuum, 
313 

Contact  potential,  309 

Contact  theories,  312 

Contact  theory  of  Voltaic  Cell, 
221 

Corpuscles,  425 

Coulomb's  Law,  45 

Coulomb's  Torsion  Balance,  87 

Crookes'  Tubes,  419 

Current,  distribution  of,  217;  unit 
of,  224;  practical  unit  of,  225; 
law  of  magnetic  force  due  to  a, 
236,  323 ;  absolute  measurement 
of,  247;  action  of  a  magnet  on, 
326 

Current  and  Quantity,  relation  be- 
tween, 185 

Currents,  Electric,  182  ;  measure- 
ment of,  185,  223 ;  tubes  of  force 
and,  185;  effects  due  to,  188; 
magnetic  action  of,  188 ;  thermal 


effects  of,   191;  chemical  action 

of,  191 
Currents,  Electric,  experiments  on, 

245;    heating  of  conductor  by, 

286 
Curve  of  magnetic  induction,  352 

Daily  Variations,  178 

Daniell's   Cell,  203;   electromotive 

force  of,  303 
Declination,  measurement  of  the, 

174 

Demagnetisation,  128 
Dielectric,  effect  of  the,  45 
Dip,  measurement  of  the,  172 
Disc  Electrometer,  the  Attracted,  90 
Distribution,  Electrical,  24 
Distribution  of  Current,  217 
Dry  Cell,  the,  202 
Dynamo  Machines,  395 

Earth,  Magnetism  of  the,  171 

Electric  Attraction,  3;  explanation 
of,  8,  9;  theory  of  Potential 
applied  to,  41 

Electric  Currents,  182;  measure- 
ment of,  185;  tubes  of  force  and, 
186;  effectsdueto,  188;  magnetic 
action  of,  188 ;  thermal  effects  of, 
191 ;  chemical  action  of,  191 ; 
experiments  on,  245;  heating  of 
conductor  by,  286 

Electric  Discharge  through  Gases, 
418 

Electric  Force,  Law  of,  45 ;  result- 
ant at  a  point,  47 

Electric  inertia,  411 

Electric  Intensity,  47 

Electric  motors,  401 

Electric  Spark,  the,  72 

Electric  Telegraph,  the,  406 

Electric  Waves,  414 

Electrical  Action,  Theories  of,  23 

Electrical  Distribution,  24 

Electrical  Energy,  291 

Electrical  Power,  291 

Electrical  Pressure,  or  Potential, 
26 ;  explanation  of,  28 

Electricity, quantity  of,  19;  thermo-, 
293 ;  observations  on,  294 


INDEX 


437 


Electritication,  4,  5;  by  induction, 
7 

Electro-chemical  equivalents,  5; 
determination  of,  250 

Electrodynamometer,  the,  367 

Electrolysis,  observations  on,  193; 
Faraday's  laws  of,  194;  trans- 
formations in,  304  •  Ionic  charge 
in,  421,  425 

Electromagnetic  Action,  328,  337; 
between  conductors,  336 

Electromagnetic  forces,  331 

Electromagnetic  machfnes,  392 

Electromagnets,  341 

Electrometer,  the  Attracted  Disc, 
90;  the  Quadrant,  94 

Electrometers  and  Electroscopes, 
87 

Electromotive  Force,  in  a  simple 
circuit,  210,  211;  of  a  Daniell's 
cell,  303 ;  standards  of,  204 

Electron  Theory,  427;  of  Matter, 
429 

Electrons,  425 ;  production  of,  427 ; 
and  galvanic  action,  428 ;  and 
magnetism,  429 

Electrophorus,  the,  74 

Electroscopes,  9-15;  condensing, 
68 ;  and  electrometers,  87 

Electrostatic  Actions,  explanation 
of,  23 

Electrostatic  Measurement  of  Po- 
tential, 93 

Electrostatic  and  Multicellular 
Voltmeters,  97 

Energy,  of  a  Charged  Conductor, 
55 ;  of  a  Charged  Condenser,  62 

Energy,  Electrical,  291 ;  needed  to 
magnetise  iron,  356 

Energy  changes  in  a  cell,  302 

Equilibrium  condition,  337 

Equipotential  Surfaces,  31,  53 

Equivalent,  electro-chemical,  195, 
250 

Experiments  with  Leyden  Jars,  68 

Experiments  with  the  Magnet- 
ometer, 157 

Experiments  on  electric  currents, 
245 

Experiments  on  batteries,  255 


Explanation  of  Electrostatic  Ac- 
tions, 23 

Faraday's  experiments  on  induc- 
tion, 373 

Faraday's  laws  of  electrolysis,  194 

Field  of  Force,  39 

Field,  Magnetic,  111 ;  tensions  and 
pressures  in,  118 

Force,  the  Field  of,  39 

Force,  Law  of,  44 

Force,  -Lines  of,  32 ;  and  equipoten- 
tial  surfaces,  34;  forms  of,  35; 
magnetic,  112;  tracing,  114; 
electromotive  in  a  circuit,  210, 
211 

Frictional  Machines,  70 

Galvanic  action,  428 
Galvanometer,  tangent,  227;  con- 
struction of,  230 
Galvanometer,  sine,  229 
Galvanometer,  resistance  of,  274 
Galvanometer,  ballistic,  279 
Galvanometer  Constant,  228 
Galvanometers,  223,  226 ;  sensitive, 

232  ;  moving  coil,  361 
Gases,  electric  discharge  through, 

418 

Gramme  ring,  the,  397 
Grove's  Cell,  202 

Holtz  Machine,  the,  82 
Hysteresis,  354 

Illustrations,  Mechanical,  40 

Inductance,  unit  of,  379 

Induction,  electrification  by,  7; 
magnetic,  111,  342;  curve  of, 
352  ;  Faraday's  experiments  on, 
373  ;  coefficient  of  mutual,  377  ; 
observations  on,  379 

Induction  Coil,  the,  389 

Inductive  Capacity,  45,  62 

Inertia,  electric,  411 

Influence,  of  Temperature,  130  ;  of 
external  fields,  371 

Influence  Machines,  77 

Insulating  Medium,  importance  of, 
30 


438 


INDEX 


Insulators,  Properties  of,  7 
Inverse     Square,    Laws    of     the, 

155 
Ionic  charge  in  electrolysis,  421, 

425 
Iron,  magnetic  force  in,  344;  energy 

needed  to  magnetise,  356 
Isoclinal  Line,  176 

Joule's  Law,  287;  applied  to  a 
battery,  290 

Kathode  Kays,  420 ;  charge  carried 
in,  422  ;  number  of  particles  in, 
424 

Keepers,  Action  of,  128 

Kelvin's  (Lord)  Ampere  Balance, 


Law,  Joule's,  287;  applied  to  a 
battery,  290;  Lena's,  378 

Law  of  Magnetic  Force,  44,  122  ; 
due  to  a  current,  236 

Law,  Ohm's,  212 ;  graphic  repre- 
sentation of,  218 ;  observations 
on,  252 

Laws,  Sine  and  Tangent,  148;  Law 
of  the  Inverse  Square,  155 

Leclanche  Cell,  the,  201 

Lenz's  Law,  378 

Leyden  Jar,  64 ;  Batteries  of,  66 ; 
experiments  with,  68 

Lines  of  Force,  32  ;  and  equipoten- 
tial  surfaces,  34  ;  forms  of,  35  ; 
magnetic,  112  ;  tracing,  114 

Local  Action,  197 

Machines,  Frictional,  70;  Plate 
Electrical,  70;  Influence,  77; 
Wimshurst's,  79 ;  Holtz,  82 ; 
Voss,  85 

Machines,  electromagnetic,  392 ; 
magneto- electric,  393;  dynamo, 
395 

Magnet,  in  a  Uniform  Field,  135 ; 
magnetic  force  due  to  a  simple, 
138 ;  forces  on  one  due  to  a 
second,  141 ;  determination  of 
the  axis  of,  168 ;  action  on  a 
current,  326 


Magnet,  rotation  about  a  current, 
339 

Magnetic  Action  of  Electric  Cur- 
rent, 188 

Magnetic  Attraction,  108;  Induc- 
tion, 111,  342 ;  curve  of,  352  ; 
Field,  111;  tensions  and  pres- 
sures in,  118;  Lines  of  Force, 
112;  Pole,  Unit,  123 

Magnetic  Batteries,  127 

Magnetic  Charge,  Total,  124 

Magnetic  Force,  due  to  a  Closed 
Cycle,  134  ;  due  to  a  simple 
magnetf  138  ;  law  of  due  to  a 
current,  2JJ1V-323 

Magnetic- Force,  Law  of,  122  ;  Ke- 
sultant,  125  ;  in  a  crevasse,  344  ; 
in  a  mass  of  iron,  344 ;  transmis- 
sion of,  414 

Magnetic  Maps,  174  ;  Storms,  179 

Magnetic  Moment,  measurement 
of  the  strength  of  a,  163 

Magnetic  Permeability ,  346  ;  mea- 
surement of,  349,  383  ;  circuit, 
347  ;  reluctance,  347 

Magnetic  Potential,  125 

Magnetic  Shells,  _325 

Magnetic  Survey,  174 

Magnetisation,  Methods  of,  126 ; 
Theory  of,  130,  855 

Magnetism,  Charges  of,  122 

Magnetism  of  the  Earth,  171 ;  se- 
cular variations  of,  178 

Magneto-electric  machines,  393 

Magnetometer,  experiments  with 
the,  157;  the  mirror,  160 

Magnets,  Natural,  105 ;  Artificial, 
106 ;  Solenoidal,  124 ;  Total  Mag- 
netic Charge  of,  124  ;  molecular, 
131 

Maps,  Magnetic,  174 

Matter,  electron  theory  of,  429 

Maximum  Strength  of  the  Pole  of 
a  Magnet,  130 

Measure  of  Potential,  49 

Measurement  of  Potential,  Electro- 
static, 93 

Measurement,  of  the  Dip,  172  ;  of 
the  Declination,  174  ;  of  electric 
currents,  185  ;  of  magnetic  per- 


INDEX 


439 


meability,    349 ;    of    alternating 

current,  371 

Measurement,  Units  of,  46 
Mechanical  Illustrations,  40 
Medium,    Insulating,    importance 

of,  30 

Methods  of  Magnetisation,  126 
Microphone,  the,  409 
Mirror  Magnetometer,  the,  160 
Molecular  Magnets,  131 
Morse  Instrument,  the,  407;  Morse 

Alphabet,  the,  408 
Motor,  transformations  of  energy 

in,  402;  starting  a,  403 
Motors,  electric,  401 
Moving  Coil  Galvanometers,  361 
Multicellular     and     Electrostatic 

Voltmeters,  9.7 

Multiple  Arc,  batteries  in,  208 
Mutual   Induction,    coefficient   of, 

377 

Natural  Magnets,  105 
Non-Conductors  and   Conductors, 
5,  6 

Observations,  on  Electrolysis,  193  ; 
on  Ohm's  Law,  252  :  on  Induc- 
tion, 379 

Ohm's  Law,  212 ;  graphic  repre- 
sentation of,  218 ;  observations 
on,  252 

Particles  in  Kathode  Kays,  424 

Peltier  effect,  the,  296 

Permeability,  magnetic,  346 ; 
measurement  of,  349,  383 

Plate  Electrical  Machine,  70 

Platinum  Thermometer,  the,  300 

Pole,  Unit  Magnetic,  123 

Pole  of  a  Magnet,  Maximum 
Strength,  130 

Potential  or  Electrical  Pressure, 
26 ;  explanation  of,  28 ;  zero 
of,  29;  analogy  between  tempera- 
ture and,  29 ;  applied  to  electrical 
attraction,  41 ;  measure  of,  49  ; 
unit  of,  51 ;  calculation  of,  51  ; 
relation  of  charge  and,  54 

Potential,    Electrostatic    Measure- 


ment of,    93 ;    magnetic,    125  ; 

contact,  314 

Potential  at  a  point  in  the  air,  99 
Potential  of  a  charged  conductor, 

53 

Potentiometer,  the,  260,  261 
Power,  Electrical,  291 
Practical  Unit  of  Current,  225 
Principles  of  Transformers,  388 
Production  of  Electrons,  427 
Proof  Plane,  the,  15,  16 
Properties  of  Insulators,  7 

Quadrant  Electrometer,  the,  94 
Quantity  of  Electricity,  19 

Kays,  Kathode,  420;  charge  carried 

in,  422 ;  number  of  particles  in, 

424  ;  Kontgen,  426 
Reactions,  Armature,  399 
Reduction  Factor,  228 
Relation  of  Charge  and  Potential,  54 
Reluctance,  magnetic,  347 
Replenisher,  the,  77 
Resistance,  unit  of,  213  ;  specific, 

216,  277  ;  of  galvanometer,  274  ; 

of  a  battery,  275 
Resistance  Boxes,  240 
Resistances,  comparison  of,  264 
Resultant  Electric  Force,  47,  48  ; 

at  a  point,  47 

Resultant  Magnetic  Force,  125 
Rontgen  Rays,  426 
Rotation  of  a  magnet,  339 

Secondary  Batteries,  205 

Secular  variations  of  the  Earth's 

magnetism,  178 
Self-induction,  378 
Sensitive  Galvanometers,  232 
Shells,  Magnetic,  325 
Shunts,  242,  364 
Shuttle  wound  Armature,  394 
Sine  Galvanometer,  229 
Sine  and  Tangent  Laws,  148 
Single  fluid  cells,  200 
Solenoid,  330 
Spark,  the  Electric,  72 
Specific  Resistance,  216,  277 
Square,  Law  of  the  Inverse,  155 


440 


INDEX 


Standards  of  electromotive  force, 

204 

Storms,  magnetic,  179 
Strength  of  the  Pole  of  a  Magnet, 

130 

Surface  Density,  24 
Surfaces,    Equipotential,    31,    53 ; 

and  lines  of  force,  34 
Survey,  Magnetic,  174 
Susceptibility,  129 

Tangent  Galvanometer,  227  ;  con- 
struction of,  230 
Tangent  Laws,  148 
Telegraph,  the  electric,  406 
Telegraphy,  wireless,  416 
Telephone,  the,  408 
Temperature,  29  ;  influence  of,  130 
Tensions  and  Pressures  in  Magnetic 

Field,  118 

Theories  of  Electrical  Action,  23 
Theory,  of  Magnetisation,  130,  355; 

Volta's,  309 
Theory    of    Potential    applied    to 

Electrical  Attraction,  41 
Thermal  effects  of  a  Current,  191 
Thermo-electricity,  293 ;    observa- 
tions on,  294 

Thermometer,  the  Platinum,  300 
Thermopile,  the,  299 
Thomson  effect,  the,  298 
Torsion  Balance,  Coulomb's,  87 
Total  Magnetic  Charge,  124 
Tracing  lines  of  Force,  114 
Transformations    in     electrolysis, 
304 ;  in  a  Voltaic  cell,  30G ;   of 
energy  in  a  motor,  402 
Transformers,  391 ;   principles  of, 
388 


Transmission  of  magnetic  force, 
414 

Tubes  of  Force  and  Electric  Cur- 
rents, 186 

Two  fluid  batteries,  202 

Uniform  Field,  Magnet  in  a,  135 ; 

measurement  of  the  strength  of 

a,  163 

Unit  of  Capacity,  285 
Unit  of  Current,  practical,  225 
Unit  of  Inductance,  379 
Unit  Magnetic  Pole,  123 
Unit  of 'Potential,  51 
Unit  of  Resistance,  213 
Units  of  Measurement,  46 

Vacuum,  contact  experiments  in, 

313 
Variations,    secular,    178;     daily, 

178 ;  annual,  178 
Voltaic  Cell,  the,  182,197  ;  chemical 

theory  of,  219 ;  contact  theory  of, 

221 ;  transformations  in,  306 
Volta's  Theory,  309 
Voltmeters,  254,  366  ;  electrostatic 

and  multicellular,  97 
Voss  Machine,  the,  85 

Water-dropping  Accumulator,  the, 

85 

Waves,  Electric,  414 
Weston  Cell,  the,  205 
Wheatstone's  Bridge,  268 
Wimshurst's  Machine,  79 
Wireless  Telegraphy,  416 

Zero  of  Potential,  29 


CAMBRIDGE:  PRINTED  BY  j.   AND  c.  F.   CI.AY,  AT  THE  UNIVERSITY  PRESS. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
This  book  is  DUE  on  the  last  date  stamped  below. 


^  tj        1  jf  *1  f 

°CT  25  1 

DEC    3    1947 

Jct'48 


50 


i  3  1957 


RE:C 


REC'D  LI 
JAN  2i  1958 


|,iA 


LD  21-100m-12,'46(A2012sl6)41 


REC'D  LD 

(RPR  2  s 


9  >63  -10PM 


ftEC'D  LD 

JU'64-*PM 


